View
222
Download
0
Embed Size (px)
Citation preview
7/29/2019 Introduction to Probaility[1]
1/40
1
Statistics
Alan D. Smith
7/29/2019 Introduction to Probaility[1]
2/40
2An Experiment ...
7/29/2019 Introduction to Probaility[1]
3/40
3An Experiment
100 Pennies
Random Toss ...
How many heads do you expect?
How many ways could they land?
What is the probability of getting all heads?What is the probability of getting 50 heads &
50 tails?
7/29/2019 Introduction to Probaility[1]
4/40
4Analysis
Consider 4 pennies
Consider 100 pennies !!!
7/29/2019 Introduction to Probaility[1]
5/40
5
Who Cares about Pennies?
Consider a situation where a manufacturer is
concerned about market share. Historically,
10% of the population uses the product. Amarket survey of 1,000 people reveals that 120
people in the survey use the product. Has
market share increased? Should we expandproduction? How certain are you?
7/29/2019 Introduction to Probaility[1]
6/40
6
Discrete Probabil i ty
Distributions
Chapter 5
7/29/2019 Introduction to Probaility[1]
7/40
7TO DEFINE THE TERMS PROBABILITY
DISTRIBUTIONAND RANDOM VARIABLE.
TO DISTINGUISH BETWEEN A DISCRETEAND
CONTINUOUSPROBABILITY DISTRIBUTION.
TO CALCULATE THE MEAN, VARIANCE, ANDSTANDARD DEVIATIONOF A DISCRETE
PROBABILITY DISTRIBUTION.
TO DESCRIBE THE CHARACTERISTICS OF THE
BINOMIAL DISTRIBUTION.
TO INTRODUCE THE POISSON DISTRIBUTION
WEEKS GOALS
7/29/2019 Introduction to Probaility[1]
8/40
8RANDOM VARIABLES Definition:Arandom var iableis a numerical
value determined by the outcome of anexperiment. (A quantity resul ting from a random
exper iment that, by chance, can assume dif ferent
values). EXAMPLE :Consider a random experiment in
which a coin is tossed three times. Let Xbe the
number of heads. Let H represent the outcome
of a Head and T the outcome of a Tail.
7/29/2019 Introduction to Probaility[1]
9/40
9EXAMPLE (Con tinued)
TTT
TTH
THT
THH
HTT
HTH
HHT
HHH
0
1
1
2
1
2
2
3Sample
SpaceNumber of
Heads = X
7/29/2019 Introduction to Probaility[1]
10/40
10PROBABIL ITY DISTRIBUTIONS Definition:Aprobabil i ty distr ibutionis a listing
of all the outcomes of an experiment and theirassociated probabilities.
7/29/2019 Introduction to Probaility[1]
11/40
11Disc rete Distr ibut ion
7/29/2019 Introduction to Probaility[1]
12/40
12CHARACTERISTICS OF APROBABIL ITY DISTRIBUTION
The probability of an outcome must always bebetween 0 and 1.
EXAMPLE:P(0 head) = 0.125, P(1 head) = 0.375,
etc. in the coin tossing experiment. The sum of the probabilities of all mutually
exclusive outcomes is always 1.
P(0) + P(1) + P(2) + P(3) = 1
7/29/2019 Introduction to Probaility[1]
13/40
13DISCRETE RANDOM VARIABLE Definition:Adiscrete random var iableis a
variable that can assume only certain clearlyseparated values resulting from a count of some
item of interest.
EXAMPLE:Let X be the number of heads whena coin is tossed 3 times. Here the values for X
are: 0, 1, 2 and 3.
EXAMPLE:Let X be the number of start-upbusinesses that fail given a starting set of 100.
The values for X are: 0, 1, , 99, 100.
7/29/2019 Introduction to Probaility[1]
14/40
14CONTINUOUS RANDOM VARIABLE Definition:Acontinuous random var iableis a
variable that can assume one of an infinitelylarge number of values (within certain
limitations).
EXAMPLE:(a) Average GPA for all Students(b) Income
(c) ROI
7/29/2019 Introduction to Probaility[1]
15/40
15THE MEAN OF A DISCRETEPROBABILITY DISTRIBUTION
Themeanreports the central locationof the data.
Themeanis the long-run averagevalue of the
random variable.
Themeanof a probability distribution is alsoreferred to as its expected value, E(X).
Themeanis aweighted average.
7/29/2019 Introduction to Probaility[1]
16/40
16MEAN o f a PROBABILITYDISTRIBUTION (con tinued)
The mean of a discrete probability distributionis computed by the formula:
wherem(Greek letter , mu) represents the meanandP(X)is the probabil i ty of the var ious outcomes
X.
m E X X P X( ) [ ( )]
7/29/2019 Introduction to Probaility[1]
17/40
17The Variance (Standard Deviat ion )of a Disc rete Probabi l i ty Distr ibu t ion
Thevariancemeasures the amount ofspread orvariationof a distribution.
Thevarianceof a discrete distribution is denoted
by the Greek letters2(sigma squared). Thestandard deviationis obtained by taking the
square rootofs2.
7/29/2019 Introduction to Probaility[1]
18/40
18The Variance (Standard Deviat ion)-
(cont inued)
The varianceof a discrete probabilitydistributionis computed from the formula:
Thestandard deviationis computed froms (s 2)1/2(sigma).
( )s m22
X P X( )
7/29/2019 Introduction to Probaility[1]
19/40
19 U-Rent-It is a car rental
agency specializing inrenting cars to families
who need an additional
car for a short period of
time. Pat Padgett,
President, has studied
her records for the last
20 weeks and reportsthe following number of
cars rented per week.
# o f C a r s
r e n t e d
W e e k s
1 0 5
1 1 6
1 2 7
1 3 2
EXAMPLE
7/29/2019 Introduction to Probaility[1]
20/40
20Does the above data qualify as a probability
distribution?Convert the number of cars rented per week to a
probability distribution. This is shown on the
next slide.
EXAMPLE (Con t inued)
7/29/2019 Introduction to Probaility[1]
21/40
21EXAMPLE (cont inued)
7/29/2019 Introduction to Probaility[1]
22/40
22Compute the mean number of cars rented per
week. The meanm = E(X)= S[XP(X)]=Compute the variance of the number of cars
rented per week.
The variance s2 S[(X- m)2P(X)]=
EXAMPLE (cont inued)
CALCULATIONS FOR IN TABLE
7/29/2019 Introduction to Probaility[1]
23/40
23CALCULATIONS FOR m IN TABLE
FORM
7/29/2019 Introduction to Probaility[1]
24/40
24CALCULATIONS FORs2i n TABLEFORM
N u m b e r
o f C a r s
R e n t e d , X
P r o b .
P ( X )( X - m ) ( X - m ) 2 ( X - m ) 2 P ( X )
1 0 0 . 2 5 1 0 - 1 1 . 3 1 . 6 9 0 . 4 2 2 51 1 0 . 3 0 1 1 - 1 1 . 3 0 . 0 9 0 . 0 2 7 0
1 2 0 . 3 5 1 2 - 1 1 . 3 0 . 4 9 0 . 1 7 1 5
1 3 0 . 1 0 1 3 - 1 1 . 3 2 . 8 9 0 . 2 8 9 0
s 2 = 0 . 9 1 3 5m = 1 1 . 3 s = 0 . 9 5 5 8 .
7/29/2019 Introduction to Probaility[1]
25/40
25B INOMIAL PROBAB IL ITY DISTRIBUTION Thebinomial distributionhas the following
characteristics:An outcome of an experiment is classified into
one of two mutually exclusive categories -
success or failure. The data collected are the results of counts.
The probability of a success stays the same for
each trial. So does the probability of failure. The trials are independent, meaning that the
outcome of one trial does not affect the outcome
of any other trial.
BINOMIAL PROBABILITY
7/29/2019 Introduction to Probaility[1]
26/40
26BINOMIAL PROBABILITYDISTRIBUTION (con tinued)
To construct a binomial distr ibution, we let
nbe the number of trials.
rbe the number of observed successes.
pbe the probability of success on each trial. qbe the probability of failure, found by q =1-p.
4! = 4*3*2*1
0! = 1! = 1
P rn
r n r prqn r( )!
!( )!
7/29/2019 Introduction to Probaility[1]
27/40
27 The Department of Labor Statistics for the state
of Kentucky reports that 20% of the workforcein Treble County is unemployed. A sample of 14
workers is obtained from the county. Compute
the following probabilities (Binomial): three are unemployed. (n= 14, p= 0.2).
P(r= 3) = ?
EXAMPLE
7/29/2019 Introduction to Probaility[1]
28/40
28P(r = 3)USING THE FORMULAGiven : (r= 3, n= 14, p= 0.2, q= 0.8)
Recall
so,
P(r= 3) =
=BINOMDIST(3, 14, 0.2, FALSE) via EXCEL
P r
n
r n r p
r
q
n r
( )
!
!( )!
7/29/2019 Introduction to Probaility[1]
29/40
29 at least one of the sampled workers is
unemployed.P(r 1) = at most two of the sampled workers are
unemployed.
P(r 2) = three or more are unemployed.
P(r 3) =
EXAMPLE (cont inued)
7/29/2019 Introduction to Probaility[1]
30/40
30Return toWho Cares about Pennies?
Consider a situation where a manufacturer is
concerned about market share. Historically,
10% of the population uses the product. Amarket survey of 1,000 people reveals that 120
people in the survey use the product. Has
market share increased? Should we expandproduction? How certain are you?
BINOMIAL PROBABILITY
7/29/2019 Introduction to Probaility[1]
31/40
31BINOMIAL PROBABILITYDISTRIBUTION
To construct a binomial distr ibution, we let
nbe the number of trials. (1000)
rbe the number of observed successes. (120 ...)
pbe the probability of success on each trial. (0.1)
qbe the probability of failure, 1-p. (0.90)
P(120) + P(121) + + P(1000) = 0.017 =
1 - P(r
7/29/2019 Introduction to Probaility[1]
32/40
32A company manufactures ball bearings to be
used on mountain bikes. It is known that 5% ofthe diameters of the bearings will be outside the
accepted limit (defective). If 6 bearings are
selected at random, what is the probability that:
Exactlyzerowil l be defective? Exactly one?
Exactly two?Exactly three?Exactly four?
Exactly five?Exactly six?
(Note: Binomial n = 6, p = 0.05)
EXAMPLE
7/29/2019 Introduction to Probaility[1]
33/40
33Note that the binomial conditions are met:
There is a constant probability of success (0.05).
There is a fixed number of trials (6).
The trials are independent. (Why?)
There are only two possible outcomes (a bearing
is either defective or nondefective).
EXAMPLE (continued)
3
7/29/2019 Introduction to Probaility[1]
34/40
34Binomial Probability Distribution for n = 6and p = 0.05
EXAMPLE (continued)
How would you use this table?
35MEAN & VARIANCE OF THE BINOMIAL
7/29/2019 Introduction to Probaility[1]
35/40
35MEAN & VARIANCE OF THE B INOMIALDISTRIBUTION
Themeanis given by
The varianceis given by
mnp
s2 1 np p( )
36EXAMPLE
7/29/2019 Introduction to Probaility[1]
36/40
36EXAMPLE For the previous example regarding defective
bearings, recall that: p= 0.05 and n= 6.
Hence, m = np = s2= np(1 - p) =
37
POISSON PROBABILITY
7/29/2019 Introduction to Probaility[1]
37/40
37
The binomial distribution of probabilities willbecome more and more skewed to the right as
the probability of success become smaller.
The limiting form of the binomial distribution
where the probability of success pis very small
andnis large is called the Poisson probabil i ty
distribution.
ThePoissondistribution can be described
mathematically using the formula:
POISSON PROBABILITY
DISTRIBUTION
38POISSON PROBABIL ITY
7/29/2019 Introduction to Probaility[1]
38/40
38
m(mu)is the arithmetic mean number of occurrences(successes) in a particular interval of time.
e is the constant 2.71828 (base of the Naperian
logarithmic system)
x number of occurrences (successes)
P(X)is the required probability.
=POISSON(x, m, FALSE) via EXCEL
POISSON PROBABIL ITY
DISTRIBUTION
P X ex
x( )
! m
m
39
EXAMPLE
7/29/2019 Introduction to Probaility[1]
39/40
39 The Maumee Urgent Care facility specializes in
caring for minor injuries, colds and flu. For theevening hours of 6 - 10 PM the mean number of
arr ivals is 4.0 per hour. Assume the arrivals
follow the Poisson distribution. Compute thefollowing probabilities.
What is the probability ofexactly 4arrivals in an
hour?
P(4) = 0.1954.
EXAMPLE
P Xe
x
x( )
!
m m
40
7/29/2019 Introduction to Probaility[1]
40/40
40
What is the probability of less than 4 arrivals in
an hour?
What is the probability of at least 4 arrivals in
an hour?
EXAMPLE (cont inued)
P Xe
x
x
( ) !
m m