© Copyright McGraw-Hill 20045-1

CHAPTER 5

Discrete ProbabilityDistributions

© Copyright McGraw-Hill 20045-2

Objectives

Construct a probability distribution for a random variable.

Find the mean, variance, and expected value for a discrete random variable.

Find the exact probability for X successes in n trials of a binomial experiment.

© Copyright McGraw-Hill 20045-3

Objectives (cont’d.)

Find the mean, variance, and standard deviation for the variable of a binomial distribution.

Find probabilities for outcomes of variables using the Poisson, hypergeometric, and multinomial distributions.

© Copyright McGraw-Hill 20045-4

Introduction

Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results.

© Copyright McGraw-Hill 20045-5

Introduction (cont’d.)

This chapter explains the concepts and applications of probability distributions. In addition, special probability distributions, such as the binomial, multinomial, Poisson, and hypergeometric distributions are explained.

© Copyright McGraw-Hill 20045-6

Random Variables

A random variable is a variable whose values are determined by chance.

© Copyright McGraw-Hill 20045-7

Discrete Probability Distribution

A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities of the values. The probabilities are determined theoretically or by observation.

© Copyright McGraw-Hill 20045-8

Calculating the Mean

In order to find the mean for a probability distribution, one must multiply each possible outcome by its corresponding probability and find the sum of the products.

)( . . .)()()( 332211 nn XPXXPXXPXXPX

© Copyright McGraw-Hill 20045-9

Rounding Rule

The mean, variance, and standard deviation should be rounded to one more decimal place than the outcome, X.

© Copyright McGraw-Hill 20045-10

Variance of a Probability Distribution

The variance of a probability distribution is found by multiplying the square of each outcome by its corresponding probability, summing those products, and subtracting the square of the mean.

The formula for calculating the variance is:

The formula for the standard deviation is:

2

2 2 2[ ( )]X P X

© Copyright McGraw-Hill 20045-11

Expected Value

Expected value or expectation is used in various types of games of chance, in insurance, and in other areas, such as decision theory.

© Copyright McGraw-Hill 20045-12

Expected Value (cont’d.)

The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable. The formula is:

The symbol E(X) is used for the expected value.

E X X P X

© Copyright McGraw-Hill 20045-13

The Binomial Distribution

Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes.

Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl, etc.

© Copyright McGraw-Hill 20045-14

The Binomial Experiment

The binomial experiment is a probability experiment that satisfies these requirements:

1. Each trial can have only two possible outcomes—success or failure.

2. There must be a fixed number of trials.

3. The outcomes of each trial must be independent of each other.

4. The probability of success must remain the same for each trial.

© Copyright McGraw-Hill 20045-15

The Binomial Experiment (cont’d.)

The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.

© Copyright McGraw-Hill 20045-16

Notation for the Binomial Distribution

P S( ) The symbol for the probability of success

The symbol for the probability of failure

The numerical probability of success

The numerical probability of failure

and

The number of trials

The number of successes

P F( )

p

q

P S p( ) P F p q( ) 1

n

X

© Copyright McGraw-Hill 20045-17

Binomial Probability Formula

In a binomial experiment, the probability of exactly X successes in n trials is

!( )

( )! !X n Xn

P X p qn X X

© Copyright McGraw-Hill 20045-18

Binomial Distribution Properties

mean

variance

standard deviation

n p

2 n p q

n p q

The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas.

© Copyright McGraw-Hill 20045-19

Other Types of Distributions

The multinomial distribution is similar to the binomial distribution but has the advantage of allowing one to compute probabilities when there are more than two outcomes.

The multinomial distribution is a general distribution, and the binomial distribution is a special case of the multinomial distribution.

© Copyright McGraw-Hill 20045-20

Poisson Distribution

The Poisson distribution is a discrete probability distribution that is useful when n is large and p is small and when the independent variables occur over a period of time.

The Poisson distribution can be used when there is a density of items distributed over a given area or volume, such as the number of defects in a given length of videotape.

© Copyright McGraw-Hill 20045-21

Formula for the Poisson Distribution

The probability of X occurrences in an interval of time, volume, area, etc., for a variable where is the mean number of occurrences per unit (area, time, volume, etc.) is

where

The letter e is a constant approximately equal to 2.7183.

P Xe

X

X( , )

!

X 0 1 2, , ,...

© Copyright McGraw-Hill 20045-22

Poisson Distribution (cont’d.)

The Poisson distribution can also be used to approximate the binomial distribution when n is large and is small (e.g., less than 5).

np

© Copyright McGraw-Hill 20045-23

Hypergeometric Distribution

When sampling is done without replacement, the binomial distribution does not give exact probabilities, since the trials are not independent. The smaller the size of the population, the less accurate the binomial probabilities will be. The hypergeometric distribution is a distribution of a variable that has two outcomes when sampling is done without replacement.

© Copyright McGraw-Hill 20045-24

Formula for the Hypergeometric Distribution

Given a population with only two types of objects (females and males, defective and nondefective, etc.) such that there are a items of one kind and b items of another kind and equals the total population, the probability of selecting without replacement a sample of size n with X items of type a and items of type b is

a bP X( )

n X

P XC C

Ca X b n X

a b n( )

( )

© Copyright McGraw-Hill 20045-25

Applications of Hypergeometric Distribution

Objects are often manufactured and shipped to a company. The company selects a few items and tests to see whether they are satisfactory or defective. The company must know the probability of getting a specific number of defects to make the decision to accept or reject the whole shipment based on a small sample. To do this, the company uses the hypergeometric distribution.

© Copyright McGraw-Hill 20045-26

Summary

A probability distribution can be graphed, and the mean, variance, and standard deviation can be found.

The mathematical expectation can also be calculated for a probability distribution.

Expectation is used in insurance and games of chance.

© Copyright McGraw-Hill 20045-27

Summary (cont’d.)

The most common probability distributions are the binomial, multinomial, Poisson, and hypergeometric distributions.

The binomial distribution is used when there are only two outcomes for an experiment, a fixed number of trials, the probability is the same for each trial, and the outcomes are independent of each other.

© Copyright McGraw-Hill 20045-28

Summary (cont’d.)

The multinomial distribution is an extension of the binomial distribution and is used when there are three or more outcomes for an experiment.

The hypergeometric distribution is used when sampling is done without replacement.

The Poisson distribution is used in special cases when independent events occur over a period of time, area, or volume.

© Copyright McGraw-Hill 20045-29

Conclusion

Many decisions in business, insurance, and other real-life situations are made by assigning probabilities to all possible outcomes pertaining to the situation and then evaluating the results.