Chapter 5

Discrete Probability Distributions

Probability Experiment

A probability experiment is any activity that produces uncertain or “random” outcomes

Random Variable

A random variable is a rule or function that translates the outcomes of a probability experiment into numbers.

Table 5.1 Illustrations of Random Variables

EXPERIMENT Possible Random Variables Type of Variable

Commuting to work Time it takes to get to workNumber of red lights on the wayAmount of gas consumed

ContinuousDiscreteContinuous

Advertising a product Number of customer responsesNumber of units sold

DiscreteDiscrete

Taking inventory Number of damaged items foundRemaining shelf life of an item

DiscreteContinuous

Playing a round of golf Driving distance off the first teeNumber of parsNumber of lost balls

ContinuousDiscreteDiscrete

Manufacturing a product Amount of waste produced (lbs.)Number of units finished in an hour

ContinuousDiscrete

Interviewing for a job Number of rejectionsDuration of the interviewElapsed time before being hired

DiscreteContinuousContinuous

Buying stocks Number of your stocks that increase in valueAmount of sleep lost from worry

DiscreteContinuous

Discrete Random Variable

A discrete random variable has separate and distinct values, with no values possible in between.

Continuous Random Variable

A continuous random variable can take on any value over a given range or interval.

Probability Distribution

A probability distribution identifies the probabilities that are assigned to all possible values of a random variable.

Producing a Discrete Probability Distribution

Step 1: Defining the Random Variable

Step 2: Identifying Values for the Random

Variable

Step 3: Assigning Probabilities to Values

of the Random Variable

Figure 5.1 Probability Tree for the Management Training Example

J (.7)

Jones Fails

S (.9)

J (.7)

S' (.1)

Smith Passes

Smith Fails

Jones Passes

S∩J

S∩J’

S'∩J

S'∩J’

2

1

1

0

.63

.27

.07

.03

Jones Fails

J' (.3)

Jones Passes

J' (.3)

(1)

(2)

(3)

(4)

Outcome x P(x)

Probability Distribution for the Training Course Illustration

Number ofManagers Passing

xProbability

P(x)

0 .03

1 .34

2 .63

1.0

Figure 5.2 Graphing the Management Training Distribution

P(x)

.6

.3

.1

Number of Managers Passing

0 1 2 x

Expected Value for a (5.1)

Discrete Probability Distribution

E(x) = x P(x)

Distribution Variance (5.2)

2 = )())(( 2 xPxEx

The Binomial Distribution

Figure 5.3 Probability Tree for the Coin

Toss Example

H (.4)

H (.4)

H (.4)

T (.6)

T (.6)

H (.4)

.096

T (.6)

H (.4).096

H (.4)

T (.6)

H (.4)

T (.6)

T (.6)

T (.6)

.144

.144

.144

.216

.0643 Heads

.096

2 Heads

2 Heads

1 Head

2 Heads

1 Head

1 Head

0 Heads

The Binomial Conditions

(1) The experiment involves a number of “trials”— that is, repetitions of the same act. We’ll use n to designate the number of trials.

(2) Only two outcomes are possible on each of the trials. This is the “bi” part of “binomial.” We’ll typically label one of the outcomes a success, the other a failure.

(3) The trials are statistically independent. Whatever happens on one trial won’t influence what happens on the next.

(4) The probability of success on any one trial remains constant

throughout the experiment. For example, if the coin in a coin-toss experiment has a 40% chance of turning up heads on the first toss, then that 40% probability must hold for every subsequent toss. The coin can’t change character during the experiment. We’ll normally use p to represent this probability of success.

The Binomial Probability Function (5.4)

n!

(n x)!x!p (1 p)x (n x)

P (x) =

Expected Value for a (5.5) Binomial Distribution

E(x) = n ּ p

Variance for a Binomial Distribution (5.6)

2 = n ּp ּ(1-p)

Standard Deviation for a (5.7) Binomial Distribution

= )1( ppn

Symmetric

Figure 5.4 Some Possible Shapes

for a Binomial Distribution

Positively Skewed

0 1 2 3 4 5 6 x

P(x)

0 1 2 3 4 5 6 x

P(x)

Negatively Skewed

0 1 2 3 4 5 6 x

P(x)

The Poisson Distribution

The Poisson Conditions

(1) We need to be assessing probability for the number of

occurrences of some event per unit time, space, or distance.

(2) The average number of occurrences per unit of time, space, or distance is constant and proportionate to the size of the unit of time, space or distance involved.

(3) Individual occurrences of the event are random and statistically independent.

Poisson Probability Function (5.8)

P(x) = !xe

x

Figure 5.5 Graphing the Poisson Distribution for = 1

mean=1

P(x)

.3

.2

.1

0 1 2 3 4 5 6 x

Figure 5.6 Matching Binomial and

Poisson Distributions

0 1 2 3 4 5 6 x

P(x) .3

.2

.1

BINOMIALn= 20, p = .10

POISSON= 2

P(x) .3

.2

.1

0 1 2 3 4 5 6 x