ECON 4630 ECON 5630

TOPIC #4: DISCRETE PROBABILITY DISTRIBUTIONSI. Discrete Probability Distributions in General

A. Definition: A random variable is a real-valued set function whose value is determined by the outcome of an experiment.

1. Examples:

2. Notation:

3. Probability Distributions in General

Outcome Probability outcomesP(X=0)

P(X=1)

P(X=2)

P(X=3)

P(X=4)

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B. Calculating the mean:

X P(x) xP(x)

0 0.0731

1 0.2700

2 0.3738

3 0.2300

4 0.0531

C. Calculating the variance:

X P(x) (x-) (x-)2 (x-)2P(x) x2P(x)

0 0.0731

1 0.2700

2 0.3738

3 0.2300

4 0.0531

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D. Rules for Transforming Random Variables

Example: Let X = number of dots that turn up on a die

x P(x) xP(x) (x-x)2P(x) x2P(x)1

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3

4

5

6

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Suppose this game has a payoff that is a linear function of X:, Specifically,

suppose Y = 2X + 8.

y P(y) yP(y) (y-x)2P(y) y2P(y)

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II. The Uniform Distribution

A. Description

B. Probability Distribution Function

Suppose a and b are the minimum and maximum values, respectively. Then:

Example #1:

x P(x)0 .11 .12 .13 .14 .15 .16 .17 .18 .19 .1Sum 1.0

What is a here?

What is b here?

What’s the probability distribution function?

Example #2: Throwing a Die

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x P(x)1 0.16672 0.16673 0.16674 0.16675 0.16676 0.1667Sum 1.0

What is a here?

What is b here?

What’s the probability distribution function?

C. Mean of a Discrete Uniform Variable

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D. Variance of a Discrete Uniform Variable

III. The Binomial (or Bernoulli) Distribution

A. Description

B. Probability Distribution Function

Let n be the number of “trials,” X be the number of “successes,” and π be the probability of “success.” Then:

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x

n

0 1 2 3 4 5 6 7 8 9 10

11 1

21 2 1

31 3 3 1

41 4 6 4 1

51 5 10 10 5 1

61 6 15 20 15 6 1

71 7 21 35 35 21 7 1

81 8 28 56 70 56 28 8 1

91 9 36 84 126 126 84 36 9 1

101 10 45 120 210 252 210 120 45 10 1

111 11 55 165 330 462 462 330 165 55 11

121 12 66 220 495 792 924 792 495 220 66

131 13 78 286 715 1,287 1,716 1,716 1,287 715 286

141 14 91 364 1,001 2,002 3,003 3,432 3,003 2,002 1,001

151 15 105 455 1,365 3,003 5,005 6,435 6,435 5,005 3,003

161 16 120 560 1,820 4,368 8,008 11,440 12,870 11,440 8,008

171 17 136 680 2,380 6,188 12,376 19,448 24,310 24,310 19,448

181 18 153 816 3,060 8,568 18,564 31,824 43,758 48,620 43,758

191 19 171 969 3,876 11,628 27,132 50,388 75,582 92,378 92,378

201 20 190 1,140 4,845 15,504 38,760 77,520 125,970 167,960 184,756

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Example #1: What is the probability of having exactly 1 girl in a 4-child family?

What’s “success”?

What’s n?

What’s π?

What’s the probability distribution function?

Does the answer change if we define “success” differently?

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Example #2: What is the probability of having exactly 3 boys in an 8-child

family?

What’s “success”?

What’s n?

What’s π?

What’s the probability distribution function?

C. Mean of a Binomial Variable

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D. Variance of a Binomial Variable

E. Examples

Example #1: # of boys in an 8-child family

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Example #2: In a recent study, 90% of homes in the U.S. were found to have color TVs. In a sample of 9 homes, what is the probability that:

a. all 9 have color TVs?b. less than 5 have color TVs?c. at least 7 have color TVs?

F. Excel commands

=BINOMDIST(S, n, π, Cumulative)

Where:S is the number of successes in n trialsN is the number of trialsπ is the probability of successCumulative is “TRUE” is you want the cumulative probability distribution and “FALSE” otherwise

For Example #2 above, if “success” means having a color TV,

1 A B C2 Number of

SuccessesProbability Cumulative Probability

3 0 =BINOMDIST(A3, 9, 0.9, FALSE) =BINOMDIST(A3, 9, 0.9, TRUE)4 1 =BINOMDIST(A4 9, 0.9, FALSE) =BINOMDIST(A4, 9, 0.9, TRUE)5 2 =BINOMDIST(A5 9, 0.9, FALSE) =BINOMDIST(A5, 9, 0.9, TRUE)6 3 =BINOMDIST(A6 9, 0.9, FALSE) =BINOMDIST(A6, 9, 0.9, TRUE)7 4 =BINOMDIST(A7 9, 0.9, FALSE) =BINOMDIST(A7, 9, 0.9, TRUE)8 etc. etc. etc.

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IV. The Hypergeometric Distribution

A. Description

B. Probability Distribution Function

Let: n = the number of “trials” S = the number of “successes” in the populationN = the population sizeN-S = the number of “failures” in the populationx = the number of “successes” in the sample Then:

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Example #1: The Economics Department at UNT consists of 10 faculty members, 6 of whom are tenured. Suppose Dr. Cobb wants to set up a committee of 4 to study ways to attract more Economics majors. Assuming he selects the committee at random,i) what is the probability that all committee members are tenured?ii) What is the probability that at least one committee member is not

tenured?

What’s “success”?

What’s n?

What’s S?

What’s N?

What’s the probability distribution function?

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Example #2 1 : According to an industry publication, Coke Classic and Pepsi ranked number one and two in sales (The Wall Street Journal Almanac, 1998). Assume that in a group of 11 individuals 7 preferred Coke Classic and 4 preferred Pepsi. A random sample of 3 of these individuals is selected.i) what is the probability that exactly 2 preferred Coke Classic?ii) What is the probability that the majority preferred Pepsi?

What’s “success”?

What’s n?

What’s S?

What’s N?

What’s the probability distribution function?

1 Adapted from Andreson et al., Statistics for Business and Economics, 9 th edition .

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C. Mean of a Hypergeometric Variable

D. Variance of a Hypergeometric Variable

E. Excel commands

=HYPGEOMDIST(X, n, S, N)

Where:X is the number of successes in the samplen is the number of trialsS is the number of successes in the populationN is the population size

For Example #2 above, if “success” means preferring Coke,

1 A B2 Number of Successes Probability3 0 =HYPGEOMDIST(A3, 3, 7, 11)4 1 =HYPGEOMDIST(A4, 3, 7, 11)5 2 =HYPGEOMDIST(A5, 3, 7, 11)6 3 =HYPGEOMDIST(A6, 3, 7, 11)

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V. The Poisson Distribution

A. Description

B. Probability Distribution Function

Let: μ = the average number of “successes” in a particular interval of time of space e = 2.71828X = the number of “successes” Then:

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Example #1: The number of typos on a page is distributed Poisson with an average of 0.8. What is the probability distribution of typos?

What’s “success”?

What’s the probability distribution function?

Example #2: Suppose the number of accidents occurring at a job site during a 4-week period is a Poisson random variable with = 2. What is the probability that there are 1 or fewer accidents during a 4-week interval?

What’s “success”?

What’s the probability distribution function?

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C. Mean of a Poisson Variable

D. Variance of a Poisson Variable

E. The Poisson Using Excel

=POISSON(x, μ, cumulative)

Where:x = number of “successes”μ = mean number of “successes”cumulative is “TRUE” for cumulative probability distribution and “FALSE” otherwise.

For Example #2 above, if “success” means AN ACCIDENT,

1 A B C2 Number of

SuccessesProbability Cumulative Probability

3 0 =POISSON(A3, 2, FALSE) =POISSON(A3, 2, TRUE)4 1 =POISSON(A3, 2, FALSE) =POISSON(A3, 2, TRUE)5 2 =POISSON(A3, 2, FALSE) =POISSON(A3, 2, TRUE)6 3 =POISSON(A3, 2, FALSE) =POISSON(A3, 2, TRUE)7 4 =POISSON(A3, 2, FALSE) =POISSON(A3, 2, TRUE)8 etc. etc. etc.

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NOTES:

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