Chapter 5 Several Discrete Distributions General Objectives: Discrete random variables are used in many practical applications. These random variables

Chapter 5 Several Discrete Distributions

General Objectives:

Discrete random variables are used in many practical applications. These random variables are often used to describe the number of occurrences of a specified or a fixed number of trials or a fixed unit of time or space.

©1998 Brooks/Cole Publishing/ITP

Specific Topics

1. The binomial probability distribution

2. The mean and variance for the binomial random variance

3. The Poisson probability distribution

4. The hypergeometric probability distribution


5.1 Introduction

Discrete random variables take on only a finite or countably infinite number of values.

Three discrete probability distributions serve as models for a large number of these applications:

- The binomial

- The Poisson

- The hypergeometric


5.2 The Binomial Probability Distribution

A coin-tossing experiment is a simple example of an important discrete random variable called the binomial random variable.

Other situations that are similar to the coin-tossing experiment:

- A sociologist is interested in the proportion of elementary school teachers who are men.

- A soft-drink marketer is interested in the proportion of cola drinkers who prefer her brand.

- A geneticist is interested in the proportion of the population who possess a gene linked to Alzheimer’s disease.


Definition: A binomial experiment is one that has these five characteristics:

1. The experiment consists of n identical results.

2. Each trial results in one of two outcomes: one outcome is called a success, S, and the other a failure, F.

3. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to (1 p) q.

4. The trials are independent.

5. We are interested in x, the number of successes observed during the n trials, for x 0, 1, 2, …, n.


See Examples 5.1 and 5.2 for examples of binomial experiments.

Example 5.1

Suppose there are approximately 1,000,000 adults in a country and an unknown proportion p favor term limits for politicians. A sample of 1000 adults will be chosen in such a way that every one of the 1,000,000 adults has an equal chance of being selected, and each adult is asked whether he or she favors term limits. (The ultimate objective of this survey is to estimate the unknown proportion p, a problem that we will discuss in Chapter 8 ). Is this a binomial experiment?

Solution

Does the experiment have the five binomial characteristics?

1. A “trial” is the choice of a single adult from the 1,000,000 adults in the country. This sample consists of n = 1000 identical trials.


2. Since each adult will either favor or not favor term limits, there are two outcomes that represent the “successes” and “failures” in the binomial experiment.

3. The probability of success, p, is the probability that an adult favors term limits. Does this probability remain the same for each adult in the sample? For all practical purposes, the answer is yes. For example, if 500,000 adults in the population favor term limits, then the probability of a “success” when the first adult is chosen is 500,000 / 1,000,000 = 1/ 2. When the second adult is chosen, the probability p changes slightly, depending on the first choice. That is, there will be either 499,999 or 500,000 successes left among the 999,999 adults. In either case, p is still approximately equal to 1/ 2.

4. The independence of the trials is guaranteed because of the large group of adults from which the sample is chosen. The probability of an adult favoring term limits does not change depending on the response of previously chosen people.

5. The random variable x is the number of adults in the sample who favor term limits.


Rule of thumb:

If the sample size is large relative to the population size — in particular, if n / N .05 — then the resulting experiment is not binomial.

The Binomial Probability Distribution:

- A binomial experiment consists of n identical trials with probability of success p on each trial.

- The probability of k successes in n trials is

for values of k 0, 1, 2,…, n.

- The symbol

where n! n(n 1)(n 2)…(2)(1) and 0! 1.


)()( ))!(/!( knkknknk qpknknqpC

)!(!knk

nCn

k

Mean and Standard Deviation for the Binomial Random Variable:

The random variable x, the number of successes in n trials, has a probability distribution with this center and spread:

Mean: np

Variance: 2 npq

Standard deviation:


npq

Cumulative binomial probabilities (CBP) are the sum of the individual binomial probabilities from 0 to some specified number k.

Use CBP tables to calculate binomial probabilities according to these steps:

- Find the necessary values of n and p and isolate the appropriate column in the CBP table.

- Find P(x k) in the line marked k and rewrite the probability you need so that it is in this form.

- List the values of x in your event. From the list, write the event as either the difference of two probabilities

P (x a ) P (x b) or 1 P (x a) See Table 5.1 for a portion of the cumulative binomial table.

See Examples 5.5 through 5.8 for examples of the use of the binomial distribution.


5.3 The Poisson Probability Distribution

The Poisson probability distribution is a good model for data that represent the number of occurrences of a specified event in a given unit of time or space.

Some examples of Poisson random variables:

- The number of calls received by a switchboard during a given period of time.

- The number of bacteria per small volume of fluid

- The number of customer arrivals at a checkout counter during a given minute

- The number of machine breakdowns during a given day

- The number of traffic accidents at a given intersection during a given time period


x represents the number of events that occur in a period of time or space during which an average of such events can be expected to occur.

The Poisson Probability Distribution:

Let be the average number of times that an event occurs in a certain period of time or space. The probability of k occurrences of this event is

for values of k 0, 1, 2, 3, – The mean and standard deviation of the Poisson random

variable are

Mean: Standard deviation:


!)(

ke

kxPk

Use cumulative Poisson tables (CPT) to calculate Poisson probabilities:

- Find the necessary values of . Isolate the appropriate column

in CPT.

- CPT gives P(x k) in the line marked k. Rewrite the probability you need so that it is in this form.

- List the values of x in your event.

- From the list, write the event as either the difference of two probabilities

P (x a) P (x b) or 1 P (x a)

See Figure 5.3 for graphs of the Poisson. See Examples 5.9 – 5.11 for an example of the Poisson.



Figure 5.3 Poisson probability distributions for = .5, 1, and 4


Example 5.9

The average number of traffic accidents on a certain section of highway is two per week. Assume that the number of accidents follows a Poisson distribution with 2.

1. Find the probability of no accidents on this section during a 1-week period.

2. Find the probability of at most three accidents on this section of highway during a 2-week period.

Solution

1. The average number of accidents per week is 2. There- fore, the probability of no accidents on this section of highway

during a given week is:

135335.!0

2)0()0( 2

20

e

epxP

2. During a 2-week period, the average of accidents on this section of highway is 2(2) 4. The probability of at most three accidents during a 2-week period is:

The Poisson probability distribution provides a simple, easy-to-compute, and accurate approximation to binomial probabilities when n is large and np is small, preferably with np 7.

An approximation suitable for larger values of np will be given in Chapter 6.


)3()2()1()0()3( ppppxP

5.4 The Hypergeometric Probability Distribution

Suppose you are selecting a sample of elements from a population and you record whether or not each element possesses a certain characteristic. You are recording the typical success or failure data found in binomial experiments.

The sample survey of Example 5.1 and the sampling for defectives of Example 5.2 are practical illustrations of these sampling situations.

If the number of elements in the population is small in relation to the sample size (n / N .05), the probability of a success for a given trial is dependent on the outcomes of preceding trials.

The number x of successes follows what is known as a hypergeometric probability distribution.


It is easy to visualize the hypergeometric random variable x by thinking of a bowl containing M red balls and N M white balls, for a total of N balls in the bowl.

You select n balls from the bowl and record x, the number if red balls that you see. If you now define a “success” to be a red ball, you have an example of the hypergeometric random variable x.

The Hypergeometric Probability Distribution: A population contains M successes and N M failures. The

probability of exactly k successes in a random sample of size n is

for values of k that depend on N, M, and n with


C

CCkxP

Nn

MNkn

mk

)(

)!(!!

nNnN

CNn

The mean and variance of a hypergeometric random variable are very similar to those of a binomial random variable with a correction for the finite population size:

See Examples 5.12 and 5.13 for illustrations of the use of the hypergeometric distribution.


1 and 2

NnN

NMN

NM

nNM

n

Key Concepts and Formulas

I. The Binomial Random Variable

1. Five characteristics: n identical independent trials, each resulting in either success S or failure F; probability of success is p and remains constant from trial to trial; and x is the number of successes in n trials.

2. Calculating binomial probabilities

a. Formula:

b. Cumulative binomial tables

c. Individual and cumulative probabilities using Minitab


knknk qpCkxP )(

3. Mean of the binomial random variable: np

4. Variance and standard deviation: 2 npq and

II. The Poisson Random Variable

1. The number of events that occur in a period of time or space, during which an average of such events are expected to occur

2. Calculating Poisson probabilities

a. Formula:

b. Cumulative Poisson tables

c. Individual and cumulative probabilities using Minitab

3. Mean of the Poisson random variable: E(x)


npq

!)(

ke

kxPk

4. Variance and standard deviation: 2 and

5. Binomial probabilities can be approximated with Poisson probabilities when np 7, using np.

III. The Hypergeometric Random Variable

1. The number of successes in a sample of size n from a finite population containing M successes and N M failures

2. Formula for the probability of k successes in n trials:


C

CCkxP

Nn

MNkn

Mk

)(

3. Mean of the hypergeometric random variable:

4. Variance and standard deviation:


NM

n

22 and 1

NnN

NMN

NM

n

Example

A survey of 50 educators is conducted to determine the proportion in favor of additional Federal funding for education. Does the survey satisfy the properties of a binomial experiment?

Solution:

Check each of the five characteristics of a binomial experiment to see if they are satisfied:

1. Are there N identical trials?

Yes, there are N = 500 trials, all the same.

2. Does each trial result in one of two outcomes?

Yes, each educator interviewed either favors or does not

favor the additional Federal funding.


3. Is the probability of success the same from trial to trial?

Yes, If we let “success” denote an educator favoring the additional funding, then assuming the list of educators from which the sample was drawn is large, the probability of success will (for all practical purposes) remain constant from trial to trial.

4. Are the trials independent?

Yes, the outcome of one interview is unaffected by the results of the other interviews.


5. Is the random variable of interest to the experimenter the number of successes Y in the sample?

Yes, we are interested in the number of educators in the

sample of 500 who favor additional Federal funding for education.

Since all five characteristics are satisfied, the survey represents a binomial experiment.


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Chapter 5 Several Discrete Distributions General Objectives: Discrete random variables are used in many practical applications. These random variables