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BU255: Statistics Exam-AID. By: Ryan Pink Some images used from course slides. Agenda. Chapter 2-8.. Go through them all.. Show you the formulas.. Use examples for each.. Answer any questions you have.. Leave you with a sick package.. Then, tell your friends to come support! . - PowerPoint PPT Presentation
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BU255 Statistics Exam-AID
By Ryan PinkSome images used from course
slides
Agenda
bull Chapter 2-8ndash Go through them allndash Show you the formulasndash Use examples for eachndash Answer any questions you havendash Leave you with a sick packagendash Then tell your friends to come support
Chapter 2
bull What is statisticsndash A way of getting information from datandash Is the science of estimating info about a
POP based on analysis from a SAMPLE
bull Population vs Samplendash POP complete setndash SAM subset of the POP
bull We make estimates or inferences about the POP from the sample data
Chapter 2
bull Parameter and Statisticsndash PARA Describes the population
ie pop mean (μ) or pop variance (σ2)
ndash STAT describes a sample an estimate of the population parameter
ie sample mean or sample variance (s2)
Chapter 2
bull Descriptive Statistics Uses data collected on a group to describe or reach conclusions on that same group
bull Inferential Statistics Uses data collected on a sample to describe or reach conclusions on the population that the sample represents
bull Types of Datandash Nominal ndash Ordinalndash Interval
Chapter 2
bull NOMINAL can only be used to classify or categorize
ndash Frequency how many times did it occurndash Relative Frequency what percentage of
the time did it occurndash Only Pie Graphs and Bar graphs
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Agenda
bull Chapter 2-8ndash Go through them allndash Show you the formulasndash Use examples for eachndash Answer any questions you havendash Leave you with a sick packagendash Then tell your friends to come support
Chapter 2
bull What is statisticsndash A way of getting information from datandash Is the science of estimating info about a
POP based on analysis from a SAMPLE
bull Population vs Samplendash POP complete setndash SAM subset of the POP
bull We make estimates or inferences about the POP from the sample data
Chapter 2
bull Parameter and Statisticsndash PARA Describes the population
ie pop mean (μ) or pop variance (σ2)
ndash STAT describes a sample an estimate of the population parameter
ie sample mean or sample variance (s2)
Chapter 2
bull Descriptive Statistics Uses data collected on a group to describe or reach conclusions on that same group
bull Inferential Statistics Uses data collected on a sample to describe or reach conclusions on the population that the sample represents
bull Types of Datandash Nominal ndash Ordinalndash Interval
Chapter 2
bull NOMINAL can only be used to classify or categorize
ndash Frequency how many times did it occurndash Relative Frequency what percentage of
the time did it occurndash Only Pie Graphs and Bar graphs
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull What is statisticsndash A way of getting information from datandash Is the science of estimating info about a
POP based on analysis from a SAMPLE
bull Population vs Samplendash POP complete setndash SAM subset of the POP
bull We make estimates or inferences about the POP from the sample data
Chapter 2
bull Parameter and Statisticsndash PARA Describes the population
ie pop mean (μ) or pop variance (σ2)
ndash STAT describes a sample an estimate of the population parameter
ie sample mean or sample variance (s2)
Chapter 2
bull Descriptive Statistics Uses data collected on a group to describe or reach conclusions on that same group
bull Inferential Statistics Uses data collected on a sample to describe or reach conclusions on the population that the sample represents
bull Types of Datandash Nominal ndash Ordinalndash Interval
Chapter 2
bull NOMINAL can only be used to classify or categorize
ndash Frequency how many times did it occurndash Relative Frequency what percentage of
the time did it occurndash Only Pie Graphs and Bar graphs
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull Parameter and Statisticsndash PARA Describes the population
ie pop mean (μ) or pop variance (σ2)
ndash STAT describes a sample an estimate of the population parameter
ie sample mean or sample variance (s2)
Chapter 2
bull Descriptive Statistics Uses data collected on a group to describe or reach conclusions on that same group
bull Inferential Statistics Uses data collected on a sample to describe or reach conclusions on the population that the sample represents
bull Types of Datandash Nominal ndash Ordinalndash Interval
Chapter 2
bull NOMINAL can only be used to classify or categorize
ndash Frequency how many times did it occurndash Relative Frequency what percentage of
the time did it occurndash Only Pie Graphs and Bar graphs
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull Descriptive Statistics Uses data collected on a group to describe or reach conclusions on that same group
bull Inferential Statistics Uses data collected on a sample to describe or reach conclusions on the population that the sample represents
bull Types of Datandash Nominal ndash Ordinalndash Interval
Chapter 2
bull NOMINAL can only be used to classify or categorize
ndash Frequency how many times did it occurndash Relative Frequency what percentage of
the time did it occurndash Only Pie Graphs and Bar graphs
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull NOMINAL can only be used to classify or categorize
ndash Frequency how many times did it occurndash Relative Frequency what percentage of
the time did it occurndash Only Pie Graphs and Bar graphs
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull ORDINAL can be used to rank or order objects
bull Nominal and Ordinal level data are referred to as nonmetric or qualitative data
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull INTERVAL distances bw numbers have meaningndash Can draw Histograms to get probability
proportionsbull Ie average daily temperature or change in stock price
ndash Skewness a distribution lacks symmetry
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
0
2
4
6
8
10
12
1 2 3 4 5
Ser ies1
Negatively Skewed Positively SkewedBIMODAL
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 2
bull INTERVALndash Relationships between two interval variables
bull SCATTER DIAGRAMndash We are interested in 1) Linearity and 2) Direction
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
1 Measure of Central Locationndash Mean Median Mode
2 Measure of Variabilityndash Range Standard Deviation Variance
Coefficient of Variation
3 Measure of Linear Relationshipndash Covariance Correlation Coefficient of
determination
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
1 Measure of Central Locationndash Arithmetic Mean
bull Only for Interval databull Simple Averagebull 1 1 1 4 4 7 7 10 30
ndash Sum = 65 n = 9 Mean = 659 = 722
ndash Medianbull Value that falls in the middle of the setbull 1 1 1 4 4 7 7 10 30
ndash Median = 4
ndash Modebull Most frequent numberbull 1 was present three times
NOTATION
N = number in POP
n = number in SAM
u = mean of POP
x = mean of SAM
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
1 Geometric Mean (diff from Arithmetic) ndash If you invested in 2006 in RIM ($70) you doubled in
2007 (to $140) and lost your shirt in 2008 (to $50) (more accurately you lost 64 in 2008 from your 2007 level)
ndash Arithmetic mean = [100 + (-64) ] 2 = 18ndash BUT WRONG (cause you went from $70 down to
$50)ndash Geometric meanndash R1 = 100 (OR 1)ndash R2 = -64 (OR -64)ndash Rg = -ndash (your annual return is a loss of 15 - DONrsquoT
MESS THIS UP)
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
2 Measures of Variabilitybull Measures spreadbull Range difference between largest and smallest but
doesnrsquot tell you anything about the points in between
bull Calculating Variability by sum of deviations does not work (since a mean of 10 with points 0 10 and 20 (10-0 10-10 and 20-10 = 0 but mean of 10 with points 91011 is MUCH tighter but still sum to 0)
VARIENCE FOR POP VARIENCE FOR SAMPLE
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
2 Standard Deviationbull Square root of the variationbull Used to compare variability in several poprsquos and
to make statements about the general shape of a dist
bull EMPIRICAL RULE 1 stdev encompasses 68 of pointsbull 2 stdevrsquos 95 and 3 997
bull CHEBYSHEFFrsquos THEOREM k stdev encompasses of points (so for 2 1-(12)^2 = 75 or 75)
bull DIFFERENCE Empirical Rule is about NORMAL distributions if NOT NORMAL (or if you donrsquot know) use Chebysheff to be safe
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
bull Examplendash If the midterm average of those who
attended an SOS session is 80 with a standard deviation of 5 marks if dist is normal what range would include 95 of all marksbull Empirical 2 stdevrsquos so 70 - 90
ndash What range would include 889 of marks if the dist was not normalbull ChebySheff 2 stdevrsquos is 75 3 is 889 (try it) bull SO a range of 65 ndash 95 would include 889 of
marks
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
3 Measure of Linear Relationshipndash Three ways to infer strength and
directionbull Covariancebull Coefficient of Correlationbull Coefficient of Determination
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
bull Covariancendash If sets are positively correlated then
positivendash If sets are negatively related then
negativendash If no real relationship then around 0
= Sxy
= σxy
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
bull Coefficient of Correlationndash Covariance says lsquowhat is the relationship
+ or ndash ndash Coeff Of Corr says lsquohow strong is that
relationship Is it really closely linked (close to 1 or -1) or is it a weak relationship (around 0)
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
bull Coefficient of Correlationndash If -1 0 or 1 you can definitely indicate the
relationship between the two (perfectly +ve etc)ndash But for all the others between you donrsquot know the
exact amount that they are affected by each other
bull Coefficient of Determinationndash measures the amount of variation in the
dependent variable that is explained by the variation in the independent variable
ndash Denoted by R2 so just square the coefficient of correlation
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
EXAMPLE
bull How much of Obamarsquos change in popularity is directly attributed to the length of SNL skits of Palin (assume normal)
Length Obama Vote
3 mins 35
4 mins 38
6 mins 42
8 mins 50
14 mins 55
END GOAL NEED Coefficient of Determination
To get that need Coefficient of Correlation
To get that need Covariance and both standard deviationrsquos
To get that need variance
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
Length
Obama Vote
3 mins
35
4 mins
38
6 mins
42
8 mins
50
14 mins
55
LEN Variance sx2
Length Mean = 35 5 = 7 mins
Obama Mean = 220 5 = 44 of votes
(-4)(-9) +(-1)(-2) +(-3)(-6) +(1)(6) +(7)(11) = 1394 = 3475
Obama Var
(9)2 +(2)2 +(6)2 +(6)2 +(11)2 = 278n-1 = 2784 = 695
Obama Variance sy2
= Sxy
VARIENCE FOR SAMPLE
(4)2 +(1)2 +(3)2 +(1)2 +(7)2 = 76n-1 = 764 = 19
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 4
Length
Obama Vote
3 mins
35
6 mins
42
4 mins
38
8 mins
50
14 mins
55
Sxy = 3475 Sx = radicS2 = radic 19 = (len stdev) = 435
Sy = radic695 = (obmam stdev) = 833
r = 3475 (435) (833) = 0959 (between -1 and 1)
We know that there is a strong relationship but since it is not 1 exactly how much of the variance is due to the length of palinrsquos skits ndash Coefficient of Determination
R2 = 09592 = 92 92 of the variation in Obamarsquos is due to the direct length of Palinrsquos skitsBOTTOM LINE See if you can get Palinrsquos skits extended by any means necessary
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
CHAPTER 5
bull 1 Data Collectionndash 1a Published datandash 1b Observational and Experimental datandash 1c Surveysndash 1d Sampling
bull 2 Sampling Methodsndash 2a Non-probability samplingndash 2b Probability Sampling
bull 3 Errorsndash 3a Sampling Errorsndash 3b Non-sampling Errors
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 5
bull Reliability and accuracy depend on the method of collection and affect the validity of the results
bull Three most popular sources1 Published data (revenue can)
bull PRIMARY = done yourself bull SECONDARY = taking from another source
2 Observational studiesbull Uncontrolled recorded of results
3 Experimental studies bull Recording of results while controlling factors
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 5
bull Survey ndash Solicit info from peoplendash Personal Phone self-administered
bull Samplingndash Why Sampling
bull Lower Costbull Impossible population sizebull Possible destructive nature of the sampling process
ndash Probability Sampling 100 random selectionndash Non-probability sampling selecting on
researchers judgment that they are representative
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 5
bull Samplingndash Three Different Types
bullSimple Random Sampling Assign numbers generate random numbers and sample
bullStratified Random Sampling classify pop into stratrsquos and then selected randomly within each (age education race province)
ndash Can get info about whole pop about relationship between stratarsquos and among each strata
bullCluster Sampling if you canrsquot get a full pop list or they are hard to question then take a cluster (GTA or people on facebook) and sample them
ndash Issue may increase sampling error due to similarities in cluster
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 5
bull Sampling Errorsndash When the distribution of the sample is not
the same as the population (means or stdev are different)bull INCREASE SAMPLE SIZE to minimize this
error
bull Non-sampling Errorndash Mistakes made in data acquisition ndash Inc sample size does NOT fix thisndash 3 types
bull Error in Data Acquisitionbull Non-Response Errorsbull Selection Bias
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 6
1 Introduction to Probability2 Assigning Probabilities3 Basic Relationships of Events4 Joint Marginal Conditional Probability5 Rules
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Probability
bull Assigning Probabilitiesndash Classical assume equally likely and independent
bull Rolling dice (16 chance)
ndash Relative Frequency assigning probabilities on experimental or historic data
bull Forecasting based on previous demand If you sold 1 computer 20 of all working days use that going forward
ndash Subjective assign on assignorrsquos judgment bull When historic measure arenrsquot good enough often used in
conjunction with benchmarks (WEATHER FORECASTING)
ndash Theoretical use known probabilitiesbull Based on a calculated probability (like arrivals at Tim
Hortonrsquos in queue theory)
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Events
bull 4 different type of events
bull Complement of an Eventbull Union of Two Eventsbull Intersection of Two Eventsbull Mutually Exclusive Events
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 6
bull Joint Probabilityndash Intersection of two eventsndash P(A and B)ndash Question Odds you passed and you
came to an SOS session P(Pass and SOS)
STATS EXAM You will pass You will fail
You came to SOS
40 5
You didnrsquot 45 10
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Chapter 6
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)ndash Question Probability that you will pass the
exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
bull Marginal Probabilityndash The summation of a particular eventndash Add up each row and column (make new rc)
ndash P(A1) = P(A1 + B1) + P(A1 + B2)
ndash Question Probability that you will pass the exam
STATS EXAM You will pass You will fail Marginal Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
bull Conditional Probabilityndash The probability of an event GIVEN another event
ndash P(A | B) = P(A B) P(B)
ndash Question Probability that you passed given you came to an SOS session
bull P(passed | attended SOS) = P(passed and came) P(attended)
bull 40 45 = 888 STATS EXAM You will pass You will fail Marginal
Prob
You came to SOS
40 5 45
You didnrsquot 45 10 55
Marginal Prob 85 15
U
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
RULES
1 No empty set2 The probability of A is 1 minus its complement3 Union is all of A + all of B subtract what they have in
common (donrsquot double count)1 If A and B are mutually exclusive (no touching of circles)
then it is just P(A) + P(B)4 Set of A is smaller or equal to set of B if A is a subset of
B
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
RULES
bull Independent Eventsndash Events A and B are independent if P(A|B) =
P(A)
ndash If there is a 30 chance that it is going to rain on your exam day
ndash Question Probability that you passed given that it rained
bull P(passed| rained) = They are independent no correlation so
bull = P(passed) = 85
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Bayes Theorem
bull Start with your initial or prior probabilities
bull You get new infobull So now with new info you calculate revised
or posterior probabilitiesbull This process is Bayes Theorem
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Bayes Theorem
bull Bayesrsquo theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space
Conditional Probability
P(Ai|B) = P(Ai)P(B|Ai)
P(B)
KEY DIFFERENCE You are just now adding up all the partitions that contain B on the bottom since you have them all split up
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Bayes Theorem
bull Example ndash Two printer cartridge companies Alamo and
Jersey ndash Alamo makes 65 of the cartridgesndash Jersey makes 35ndash Alamo has a defective rate of 8ndash Jersey has a defective rate of 12a) Customer purchases a cartridge prob that Alamo
made it
- Cartridge is tested and it is defective (new info)
b) What is the probability that Alamo made the cartridge c) What is the probability that Jersey made the cartridge
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
ANSWER
bull The knowledge of the producer breakdown is the prior probabilityndash Alamo = 65 P(E1)
ndash Jersey = 35 P(E2)
bull We know the conditional probabilities of the defective ratesndash Alamo = 8 P(D|E1)
ndash Jersey = 12 P(D|E2)
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
ANSWER 1 TABLE
Prior Conditional Joint Posterior
Alamo 65 08 052 052094 = 553
Jersey 35 12 042 042094 = 447
Total defective
094 1000
Odds of getting an alamo cartridge that is defective if you bought it at futureshop by random
Given that you got a defective cartridge since there is a 94 chance of getting a defective one and 52 of that 94 is Alamorsquos then you have a 553 of it being Alamorsquos
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
ANSWER 2 TREE
Alamo65
Jersey35
Defective08
Defective12
Acceptable88
Acceptable92
052
598
042
308
094
Revised Probabilty Alamo = 052 094 = 553Revised Probabilty Jersey = 042 094 = 447
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
ANSWER 3 FORMULA
bull Chance defective will be an Alamo
Probably of an Alamo(65)
Probably of defective given an Alamo (08)
The summation of all the cartridge types their defective probability (find out in total how many defective ones are there) = (094)
P(Alamo) P(D|Alamo)
P(Alamo)(P(D|Alamo) + P(Jersey) P(D|Jersey)
P(Alamo | D) =
P(Alamo | D) = 65 08 (6508) + (3512)P(Alamo | D) = 052 094 = 553
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
CHAPTER 7
1 Random Variables and Probabilityndash Distributions Introduction
2 Discrete Probability Distributionsndash A Introductionndash B Mean and Variancendash C Laws of Mean and Variance
3 Bivariate Distributionsndash A Introduction Marginal probability distributionndash B Mean Variance covariance coefficient of
correlationndash C Conditional probability independencendash D Laws of summation
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Random Variable
bull Random variable definition a variable that
contains the outcomes of a chance experiment bull Two types
ndash Discrete Random Variablebull Countable number of values (students in a
class)
ndash Continuous Random Variablebull Takes on an uncountable number of possible
outcomesndash Time in 100m sprint (could be 95s or 951s or
9519shellip)
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Discrete Prob Distributions
bull Table Graph that lists all the outcomes and their probabilities = Discrete Prob Dist
bull You can calculate the prob of a certain outcomendash P(x) ndash RULESndash P(x) MUST be between 0 and 1ndash Sum of all P(xi) = 1
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Continuous Prob Distribution
bull This represents a population (since infinite amount of outcomes) and need to calculate parameters to depict distribution
ndash Need Pop mean and Pop variancendash Population Mean
bull (using discrete variables to determine parameter about pop)
ndash Population Variance bull (using discrete variables to determine parameter about
pop) OR
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Example
bull With dicendash What is the probability distributionndash What is the mean
What is the variancendash What is the stdev
Mean = 1 (16) + 2 (16) + 3(16) + 4(16) + 5(16) + 6(16) = 35
Variance = [12(16)+ 22 (16) + 32 (16) + 42 (16) + 52 (16) + 62 (16) ] ndash 352
Variance = 15166 ndash 1225 = 291
Standard Deviation = sqrt (291) = 170
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Laws Of Expected Value Var
Example Long Distance phone company bills customers 50 cents per call plus 2 cents a minute A statistician determined the mean and variance of call length to be 10 minutes and 9 minutes2 Determine the mean and variance for each call
E(cX + c) = E(cX) + c
E(cX) + c = cE(X) + c
= 2E(X) + 50
= 210 + 50
E(X) = 70 cents
V(cX + c) = V(cX)
V(cX) = c2 V(X)
= 22 (9)
V(X) = 36
Stdev(X) = 6 cents
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Bivariate (joint) Distributions
bull This is used when the relationship between 2 variables is studied
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Example
Store sells Refrigerators and Stoves the probability of selling X stoves and refrigerators in 1 day is below
Q1 What is the marginal distribution of the number of stoves sold in 1 day
Q2 What is the marginal distribution of the number of fridges sold in 1 day
Q3 What is the mean variance stdev for number of fridges sold
343927
294923 1
E(X) = 023 + 149+ 229 = 107 = mean
V(X) = (0223 + 1249 + 2226 ) ndash 1072 = 3851
Stdev(X) = sqrt (3851) = 62 fridges
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Covariance and Correlation
bull To describe the relationship between 2 random variables we use covariance and coefficient of correlation
Correlation
Covariance
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
bull Compute Correlation and Covariance
Covariance (0008 + 0114 + 0212 + 1009 + 1117 + 1213 + 2005 + 2118 + 2204 ) ndash 10793
Mean for Fridges = 107
Mean for Stoves = 093
= (1117 + 1213 + 2118 + 2204 ) ndash 10793
= 95 - 9951
= - 004 (we know the relationship is negative but not sure of how strong)
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
bull Compute Correlation and CovarianceMean for Fridges = 107
Mean for Stoves = 093
Variance for Fridges = 3851
Variance for Stoves = 1005
Covariance = -004
= -04 radic38512 radic1005
= -0643
Covariance showed us that there IS a negative relationship Coefficient of Correlation told us that it is a weak relationship (between -1 and +1)
MC NOTE If two variables are independent the covariance is zero and so is the correlation coefficient
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Conditional Probabilities (revisited)
Already did this in previous chapter but can be applied here as well
Probability that they will sell 2 stoves if only 1 fridge has been sold
343927
294923
P(S2|R1) = P(S2 and R1) P(R1)
P(S2|R1) = 18 49
= 367
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Bivariate Distribution Finance App
2 Applicationbull 1048698 3 Binomial Distributionbull 1048698 4 Poisson Distribution
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
bull MC potential question that gives you variables for mean variance of the return of two stocks and yoursquoll have to compute Expected Return and Variancendash You want to diversify your portfolio by having equal amount
of these two stocksbull RIM return mean = 11 variance = 20bull GE return mean = 06 variance = 03bull P (rho) = 1
Bivariate Distribution Finance App
E(R) = 5(11) + 5(06)
E(R) = 085 (85 return)
V(R) = 522 + 5203 + 2(5)(5)(1)(44)(17) = 059
Stdev = 2446 (24 up or down)
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binomial Distribution
bull Is a discrete probability distributionndash There is lsquonrsquo trials (fixed and finite)ndash Each result is either a success or failure (only 2
choices)ndash Probability for success is the same for each trialndash All trials are independent so past results DO not affect
future ones
bull Examplesndash Coin flippingndash Pass failndash Elections (got the vote or didnrsquot)
Number of successes
Height = probability
Sum of all bars = 1
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binomial Distribution
All you need is the probability of success (or failure) how many trials and how many you want to test for
Eg coin flip done 5 times What is the prob of 3 being heads P(X = 3)
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binomial Table
Q1 Coin flip done 5 times What is the prob of 3 being heads P(X = 3)
P(X lt= 3) ndash P(Xlt= 2)
= 813 - 500
= 313 or 313
P(x=0123) ndash P(x=012)
All cancel out except P(x=3)
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binomial Table
Q1 Oscars nominated 6 years in a row (5 nominees each year) What is the prob of you winning it more then 2 times P(X gt 2)
= 1- P(X lt= 2)
= 1 - 901
= 0099 or 99
Assume all 5 nominees have the same prob of success(2)
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binomial Table
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
= P(xlt=2) ndash P(xlt=1)
= 991 - 919
= 72
Prob is (1 in 10) or 01
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binomial Calculation
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
P(2) = 5 (2)(3) 12 (9)3
P(2) = 120 12 0010729
P(2) = 00729 = 729 or 73 as per table
Or
=binomdist(2510)
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Binom Mean and Variance
Q1 Radio phone-in competition 10 callers all trying to get on at the same time you try for the same competition 5 days this week What is the probability of you getting on twice P(x=2)
E(X) = 5(1) = 5
V(X) = 5(1)(9) = 45
STdev = radic(519) = radic45 = 67
YOU SHOULD EXPECT TO GET ON 5 +- 67 times (so between 0 and 117 times)
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Poisson Distribution
bull Poisson is to describe the distribution of the probability that lsquoxrsquo occurrences in an interval (number of successes)
bull Discrete distribution bull Examples
ndash Number of cars arriving in an hour of time ndash Number of errors on 100 pages in a textbook
bull ALL have an interval and the items are success or failure
Diff between Poisson and Binomial
No given number of trials (n) with Poisson
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Poisson
Method 1 Calculation
Method 2 Excel
Method 3 Table
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
EXAMPLE FORMULA
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
F(5) = 35271-3 5
F(5) = 12209 120
F(5) = 10
You need to add up F(0) + F(1)
F(0) = 30271-3 0
F(0) = 00502 1
F(0) = 502
F(1) = 31271-3 1
F(1) = 01507 1
F(1) = 1507
ANSWER0502 + 1507 = 2009
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
EXAMPLE EXCEL
Greg is known to have a potty mouth and some eager second year stats students counted that he swore on average 3 times per an hour of SOS Exam-AID review sessions and the distribution was poisson distributed
Q1 What is the probability that he drops 5 f-bombs a hour
Q2 What is the probability that he drops fewer then 2 bombs F(x lt 2)
=POISSION(131) CUMULATIVE prob for less then 2 successes with mean of 3 (so up to 1 cumulative)
=POISSION(530) POINT prob for 5 successes with mean of 3
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Continuous Distribution
bull Introduction to Continuous Probability Distributions
bull Uniform Distributionbull Normal Distributionbull Calculating Normal Probabilities
ndash Tablendash Excel
bull Exponential Distribution
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
CONTINUOUS FUNCTION
bull Graph of a non-discrete typendash Time speed height
bull Since no discrete values probability of a single value is 0 (Point probability)
bull You must find the area of a range ndash (so those who finished the race between 10 sec and 12 sec)
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
bull Uniform Probability Distribution
Question Gas demand is uniformly distributed between 3000 and 5000 gallons a day for Canadian gas stations What is the mean
MEAN 5000 + 3000 2 = 4000 gallons a day
VARIANCE (5000 ndash 3000)2 12 = 333333
STDEV radic333333 = 577 gallons a day
Mean (b+a)2
Variance (b - a )212
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
NORMAL DIST
To shift the curve change the mean (the middle point)
To widen the curve increase the stdev (to tighten decrease it)
Symmetric around the mean lsquoursquo and fully defined by the mean and stdev
The Bell curve
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
STANDARD NORMAL
bull When a normal curve has a mean of 0 and standard deviation of 1 it is the standard normal distribution
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
STANDARD NORMAL
All curves can be converted to standard normal
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Z-values
bull Z-score is the number of standard deviations a value (x) is above or below the mean
bull Big area is finding area under the curves
ndash Z025 = what is the z value that has an area to the wing of 25
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Normal Example
bull The GMAT is largely used as an entrance requirement to graduate business schools
bull Assuming scores are normally distributedhellip
bull Mean score was 494 std dev was 100 a) Probability a randomly selected score is between the mean and
600b) Probability of a score being over 700c) Probability a score is 550 or less d) A score is between 300 and 600 e) A score is between 350 and 450
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Normal Example
bull Answers a) 3554b) 0197c) 7123d) 8292e) 2551
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Exponential Dist
bull Continuous dist functionbull X gt= 0 (no negatives)bull All depends on the value of lamba (λ)
ndash Smaller values flatten the curve
bull Calculate Probabilityrsquos width
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Exponential Question
Q Inspection at this border crossing takes an average of 5 minutes (Exp dist)
1 What is the ldquoservice raterdquo (per hour)bull λ = 605 = 12 per hour
2 What is the probability an inspection will take precisely 5 minutesbull 0 Continuous probability
3 What is the probability an inspection will take more than 10 minutesbull P(X gt 10) = e -12(1060) = e -2 = 1353 = 13 (WATCH UNITS)
4 What is the probability an inspection will take between 25 and 75 minutes1 P(25 lt X lt 75) = e -12(2560) ndash e -12(7560) = e -05 ndash e -15
= 606 - 223 = 3438 or 3438
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
DISTRIBUTIONS
bull Binomial (table formula excel)bull Normal (table formula excel)bull Poisson (formula table excel)bull Continuous probability (formula)bull Uniform (formula)bull Exponential Distribution (table formula
excel)
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Name that Distribution
a) 6 people arrive at the ER in an hourb) Probability less than 10 minutes
elapse between arrivals at the ERc) Rolling a 4 on a 6 sided did) Number of times a Chrysler breaks
down in a yeare) Finding 7 or more left handed people
in an SOS session of 100 people
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
Answers
a) Poissonb) Exponentialc) Binomiald) Poissone) Binomial
DONE
Good Luck
DONE
Good Luck
