CHAPTER 16

Random Variables and Probability Distributions

Streamline Treatment of Probability

Sample spaces and events are good starting points for probability

Sample spaces and events become quite cumbersome when applied to real-life business-related processes

Random variables allow us to apply probability, risk and uncertainty to meaningful business-related situations

Bring Together Numerical Summaries of Data and Probability

In previous chapters we saw that data could be graphically and numerically summarized in terms of midpoints, spreads, outliers, etc.

In basic probability we saw how probabilities could be assigned to outcomes of an experiment. Now we bring them together

First: Two Quick Examples 1. Hardee’s vs. The Colonel

Hardee’s vs The ColonelOut of 100 taste-testers, 63 preferred

Hardee’s fried chicken, 37 preferred KFCEvidence that Hardee’s is better? A

landslide?What if there is no difference in the

chicken? (p=1/2, flip a fair coin)Is 63 heads out of 100 tosses that

unusual?

Example 2.Mothers Identify Newborns

Mothers Identify NewbornsAfter spending 1 hour with their newborns,

blindfolded and nose-covered mothers were asked to choose their child from 3 sleeping babies by feeling the backs of the babies’ hands

22 of 32 women (69%) selected their own newborn

“far better than 33% one would expect…”Is it possible the mothers are guessing?Can we quantify “far better”?

Graphically and Numerically Summarize a Random

ExperimentPrincipal vehicle by which we do this:random variablesA random variable assigns a number

to each outcome of an experiment

Random VariablesDefinition:A random variable is a numerical-

valued function defined on the outcomes of an experiment

S Number lineRandom variable

ExamplesS = {HH, TH, HT, TT}the random variable: x = # of heads in 2 tosses of a coinPossible values of x = 0, 1, 2

Two Types of Random Variables

Discrete: random variables that have a finite or countably infinite number of possible values

Test: for any given value of the random variable, you can designate the next largest or next smallest value of the random variable

Examples: Discrete rv’sNumber of girls in a 5 child familyNumber of customers that use an ATM

in a 1-hour period.Number of tosses of a fair coin that is

required until you get 3 heads in a row (note that this discrete random variable has a countably infinite number of possible values: x=3, 4, 5, 6, 7, . . .)

Two types (cont.)Continuous: a random variable that

can take on all possible values in an interval of numbers

Test: given a particular value of the random variable, you cannot designate the next largest or next smallest value

Which is it, Discrete or Continuous?

Discrete random variables “count”Continuous random variables

“measure” (length, width, height, area, volume, distance, time, etc.)

Examples: continuous rv’sThe time it takes to run the 100 yard

dash (measure)The time between arrivals at an ATM

machine (measure)Time spent waiting in line at the

“express” checkout at the grocery store (the probability is 1 that the person in front of you is buying a loaf of bread with a third party check drawn on a Hungarian bank) (measure)

Examples: cont. rv’s (cont.)The length of a precision-engineered

magnesium rod (measure)The area of a silicon wafer for a

computer chip coming off a production line (measure)

Classify as discrete or continuous

a x=the number of customers who enter a particular bank during the noon hour on a particular day

a discrete x={0, 1, 2, 3, …}b x=time (in seconds) required for a

teller to serve a bank customerb continuous x>0

Classify (cont.)c x=the distance (in miles) between a

randomly selected home in a community and the nearest pharmacy

c continuous x>0d x=the diameter of precision-engineered

“5 inch diameter” ball bearings coming off an assembly line

d continuous; range could be {4.5x<5.5}

Classify (cont.)

e x=the number of tosses of a fair coin required to observe at least 3 heads in succession

e discrete x=3, 4, 5, ...

CUSIP IND CONAME PE NPM60855410 4 MOLEX INC 24.7 8.740262810 5 GULFMARK INTL INC 21.4 8.181180410 4 SEAGATE TECHNOLOGY 21.3 2.246489010 9 ISOMEDIX INC 25.2 21.169318010 9 PCA INTERNATIONAL INC 21.4 4.726157010 7 DRESS BARN INC 24.5 4.590249410 4 TYSON FOODS INC 20.9 3.94886910 5 ATLANTIC SOUTHEAST AIRLINES 20.1 15.787183910 9 SYSTEM SOFTWARE ASSOC INC 23.7 11.662475210 4 MUELLER (PAUL) CO 14.5 3.936473510 7 GANTOS INC 15.7 1.800755P10 9 ADVANTAGE HEALTH CORP 23.3 5.323935910 2 DAWSON GEOPHYSICAL CO 14.9 9.368555910 4 ORBIT INTERNATIONAL CP 15.0 3.016278010 4 CHECK TECHNOLOGY CORP 17.1 3.251460610 4 LANCE INC 19.0 8.54523710 4 ASPECT TELECOMMUNICATIONS 25.7 8.274555310 4 PULASKI FURNITURE CORP 22.0 2.180819410 4 SCHULMAN (A.) INC 19.4 6.019770920 9 COLUMBIA HOSPITAL CORP 18.3 3.123790310 4 DATA MEASUREMENT CORP 11.3 2.611457710 4 BROOKTREE CORP 13.8 13.600431L10 9 ACCESS HEALTH MARKETING INC 22.4 11.029605610 4 ESCALADE INC 10.8 2.023303110 4 DBA SYSTEMS INC 6.3 5.064124610 4 NEUTROGENA CORP 27.2 9.059492810 6 MICROAGE INC 9.0 0.522821010 7 CROWN BOOKS CORP 24.4 1.8190710 4 AST RESEARCH INC 9.7 7.346978310 6 JACO ELECTRONICS INC 31.9 0.4531320 4 ADAC LABORATORIES 18.5 10.649766010 4 KIRSCHNER MEDICAL CORP 33.0 0.830205210 4 EXIDE ELECTRS GROUP INC 29.0 2.446065P10 5 INTERPROVINCIAL PIPE LN 11.9 19.219247910 4 COHERENT INC 40.2 1.2

CUSIP IND CONAME PE NPM60855410 4 MOLEX INC 24.7 8.740262810 5 GULFMARK INTL INC 21.4 8.181180410 4 SEAGATE TECHNOLOGY 21.3 2.246489010 9 ISOMEDIX INC 25.2 21.169318010 9 PCA INTERNATIONAL INC 21.4 4.726157010 7 DRESS BARN INC 24.5 4.5

Data Variables and Data DistributionsData Variables and Data Distributions

Data variables areknown outcomes.

CUSIP IND CONAME PE NPM60855410 4 MOLEX INC 24.7 8.740262810 5 GULFMARK INTL INC 21.4 8.181180410 4 SEAGATE TECHNOLOGY 21.3 2.246489010 9 ISOMEDIX INC 25.2 21.169318010 9 PCA INTERNATIONAL INC 21.4 4.726157010 7 DRESS BARN INC 24.5 4.590249410 4 TYSON FOODS INC 20.9 3.94886910 5 ATLANTIC SOUTHEAST AIRLINES 20.1 15.787183910 9 SYSTEM SOFTWARE ASSOC INC 23.7 11.662475210 4 MUELLER (PAUL) CO 14.5 3.936473510 7 GANTOS INC 15.7 1.800755P10 9 ADVANTAGE HEALTH CORP 23.3 5.323935910 2 DAWSON GEOPHYSICAL CO 14.9 9.368555910 4 ORBIT INTERNATIONAL CP 15.0 3.016278010 4 CHECK TECHNOLOGY CORP 17.1 3.251460610 4 LANCE INC 19.0 8.54523710 4 ASPECT TELECOMMUNICATIONS 25.7 8.274555310 4 PULASKI FURNITURE CORP 22.0 2.180819410 4 SCHULMAN (A.) INC 19.4 6.019770920 9 COLUMBIA HOSPITAL CORP 18.3 3.123790310 4 DATA MEASUREMENT CORP 11.3 2.611457710 4 BROOKTREE CORP 13.8 13.600431L10 9 ACCESS HEALTH MARKETING INC 22.4 11.029605610 4 ESCALADE INC 10.8 2.023303110 4 DBA SYSTEMS INC 6.3 5.064124610 4 NEUTROGENA CORP 27.2 9.059492810 6 MICROAGE INC 9.0 0.522821010 7 CROWN BOOKS CORP 24.4 1.8190710 4 AST RESEARCH INC 9.7 7.346978310 6 JACO ELECTRONICS INC 31.9 0.4531320 4 ADAC LABORATORIES 18.5 10.649766010 4 KIRSCHNER MEDICAL CORP 33.0 0.830205210 4 EXIDE ELECTRS GROUP INC 29.0 2.446065P10 5 INTERPROVINCIAL PIPE LN 11.9 19.219247910 4 COHERENT INC 40.2 1.2

DATA DISTRIBUTIONDATA DISTRIBUTIONPrice-Earnings RatiosPrice-Earnings Ratios

|||| ||||

Class(bin)

ClassBoundary Tally Frequency

1 6.00-12.99 |||| | 6 6/35 = 0.171

2 13.00-19.99 10

3 20.00-26.99 |||| |||| |||| 14

4 27.00-33.99 |||| 4

5 34.00-40.99 | 1 1/35 = 0.029

RelativeFrequency

4/35 = 0.114

14/35 = 0.400

10/35 = 0.286

Handout 2.1, P. 10

CUSIP IND CONAME PE NPM60855410 4 MOLEX INC 24.7 8.740262810 5 GULFMARK INTL INC 21.4 8.181180410 4 SEAGATE TECHNOLOGY 21.3 2.246489010 9 ISOMEDIX INC 25.2 21.169318010 9 PCA INTERNATIONAL INC 21.4 4.726157010 7 DRESS BARN INC 24.5 4.5

Data Variables and Data DistributonsData Variables and Data Distributons

Data variables areknown outcomes.

Data distributionstell us what happened.

Random Variables and Probability Distributions

Random variables areunknown chance outcomes.

Probability distributionstell us what is likely

to happen.

Data variables areknown outcomes.

Data distributionstell us what happened.

Great

Good

EconomicScenario

Profit($ Millions)

5

10

Profit Scenarios

Handout 4.1, P. 3

Random variables areunknown chance outcomes.

Probability distributionstell us what is likely

to happen.

Great

Good

OK

EconomicScenario

Profit($ Millions)

5

1

-4Lousy

10

Profit Scenarios

ProbabilityProbability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

The proportion of the time an outcome is expected to happen.

Probability DistributionProbability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

Shows all possible values of a random variable and the probability associated with each outcome.

X = the random variable (profits)xi = outcome i

x1 = 10x2 = 5x3 = 1x4 = -4

NotationProbability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

x4

X

x1

x2

x3

P is the probabilityp(xi)= Pr(X = xi) is the probability of X being

outcome xi

p(x1) = Pr(X = 10) = .20p(x2) = Pr(X = 5) = .40p(x3) = Pr(X = 1) = .25p(x4) = Pr(X = -4) = .15

NotationProbability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

Pr(X=x4)

X

Pr(X=x1)

Pr(X=x2)

Pr(X=x3)

x1

x2

x3

x4

What are the chances?

What are the chances that profits will be less than $5 million in 2009?

P(X < 5) = P(X = 1 or X = -4)= P(X = 1) + P(X = -4)= .25 + .15 = .40

Probability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

X

x1

x2

x3

x4

What are the chances?

What are the chances that profits will be less than $5 million in 2009 and less than $5 million in 2010?

P(X < 5 in 2009 and X < 5 in 2010) = P(X < 5)·P(X < 5) = .40·.40 = .16

P(X < 5) = .40

Probability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

p(x4)

X

x1

x2

x3

x4

P

p(x1)

p(x2)

p(x3)

.05

.10

.15

.40

.20

.25

.30

.35

Probability Probability HistogramHistogram

-4 -2 0 2 4 6 8 10 12

Profit

Probability

.05

.10

.15

.40

.20

.25

.30

.35

Probability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

p(x4)

X

x1

x2

x3

x4

p(x1)

p(x2)

p(x3)

.05

.10

.15

.40

.20

.25

.30

.35

Probability Probability HistogramHistogram

-4 -2 0 2 4 6 8 10 12

Profit

Probability

Lousy

OK

Good

Great

.05

.10

.15

.40

.20

.25

.30

.35

Probability

Great 0.20

Good 0.40

OK 0.25

EconomicScenario

Profit($ Millions)

5

1

-4Lousy 0.15

10

p(x4)

X

x1

x2

x3

x4

P

p(x1)

p(x2)

p(x3)

Probability distributions: requirements

Notation: p(x)= Pr(X = x) is the probability that the random variable X has value x

Requirements1. 0 p(x) 1 for all values x of X

2. all x p(x) = 1

Examplex p(x)0 .201 .902 -.10

property 1) violated:p(2) = -.10

x p(x)-2 .3-1 .31 .32 .3

property 2) violated:p(x) = 1.2

Example (cont.)

x p(x)-1 .250 .651 .10

OK 1) satisfied: 0 p(x) 1 for all x

2) satisfied: all x p(x) = .25+.65+.10 = 1

Example20% of light bulbs last at least 800 hrs;

you have just purchased 2 light bulbs.X=number of the 2 bulbs that last at

least 800 hrs (possible values of x: 0, 1, 2)

Find the probability distribution of XS: bulb lasts at least 800 hrsF: bulb fails to last 800 hrsP(S) = .2; P(F) = .8

Example (cont.)

Possible outcomes P(outcome) x(S,S) (.2)(.2)=.04 2(S,F) (.2)(.8)=.16 1(F,S) (.8)(.2)=.16 1(F,F) (.8)(.8)=.64 0

probability x 0 1 2distribution of x: p(x) .64 .32 .04

Example3 child family;X=#of boysM: child is male

P(M)=1/2(0.5121; from .5134)F: child is female

P(F)=1/2(0.4879)

Outcomes P(outcome) xMMM (1/2)3=1/8 3MMF 1/8 2MFM 1/8 2FMM 1/8 2MFF 1/8 1FMF 1/8 1FFM 1/8 1FFF 1/8 0

Probability Distribution of x

x 0 1 2 3p(x) 1/8 3/8 3/8 1/8Probability of at least 1 boy:P(x 1)= 3/8 + 3/8 +1/8 = 7/8Probability of no boys or 1 boy:p(0) + p(1)= 1/8 + 3/8 = 4/8 = 1/2

Two More Examples1. X = # of games played in a randomly

selected World SeriesPossible values of X are x=4, 5, 6, 7

2. Y=score on 13th hole (par 5) at Augusta National golf course for a randomly selected golfer on day 1 of 2011 Masters

y=3, 4, 5, 6, 7

Probability Distribution Of Number of Games Played in Randomly Selected World Series

Estimate based on results from 1946 to 2010.

x 4 5 6 7p(x) 12/65=0.18

512/65=0.18

514/65=0.21

527/65=0.4

15

Probability Histogram

Probability Distribution Of Score on 13th hole (par 5) at Augusta National Golf Course on Day 1 of 2011 Masters

y 3 4 5 6 7p(x) 0.040 0.414 0.465 0.051 0.030

Probability Histogram