Lecture Unit 4Section 4.2
Random Variables and Probability Models:
Binomial, Geometric and Poisson 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 Lecture Unit 2, and Section 4.1
In Lecture Unit 2 we saw that data could be graphically and numerically summarized in terms of midpoints, spreads, outliers, etc.
In Section 4.1 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 Colonel
Out of 100 taste-testers, 63 preferred Hardee’s fried chicken, 37 preferred KFC
Evidence 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 Variables
Definition:A random variable is a numerical-
valued function defined on the outcomes of an experiment
S Number lineRandom variable
Examples
S = {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’s
Number 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 are
known 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 are
known outcomes.
Data distributions
tell us what happened.
Random Variables and Probability Distributions
Random variables areunknown chance outcomes.
Probability distributionstell us what is likely
to happen.
Data variables are
known outcomes.
Data distributions
tell 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
Probability
Probability
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 Distribution
Probability
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 = 10
x2 = 5
x3 = 1
x4 = -4
Notation
Probability
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) = .20
p(x2) = Pr(X = 5) = .40
p(x3) = Pr(X = 1) = .25
p(x4) = Pr(X = -4) = .15
Notation
Probability
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 $5 million in 2013?
P(X < 5) = P(X = 1 or X = -4)= P(X = 1) + P(X = -4)= .25 + .15 = .40
Probability
Great 0.20
Good 0.40
OK 0.25
EconomicScenario
Profit($ 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 2013 and less than $5 million in 2014?
P(X < 5 in 2013 and X < 5 in 2014)
= P(X < 5)·P(X < 5) = .40·.40 = .16
P(X < 5) = .40
Probability
Great 0.20
Good 0.40
OK 0.25
EconomicScenario
Profit($ 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
Example
x 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
Example: light bulbs20% 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
S
S - SS
F - SF
S - FS
F - FF
F
Example: 3-child family
3 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