Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

McGraw-Hill/Irwin

Chapter 5

Discrete Random Variables

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Chapter Outline

5.1 Two Types of Random Variables5.2 Discrete Probability Distributions5.3 The Binomial Distribution5.4 The Poisson Distribution

(Optional)

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5.1 Two Types of Random Variables

Random variable: a variable that assumes numerical values that are determined by the outcome of an experiment Discrete Continuous

Discrete random variable: Possible values can be counted or listed The number of defective units in a batch of 20 A rating on a scale of 1 to 5

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Random Variables Continued

Continuous random variable: May assume any numerical value in one or more intervals The waiting time for a credit card

authorizationThe interest rate charged on a

business loan

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5.2 Discrete Probability Distributions

The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assumeNotation: Denote the values of the

random variable by x and the value’s associated probability by p(x)

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Discrete Probability Distribution Properties

1

xof each valuefor 0

all

x

xp

xp

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Expected Value of a Discrete Random Variable

The mean or expected value of a discrete random variable X is:

is the value expected to occur in the long run and on average

xAll

X xpx

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Variance

The variance is the average of the squared deviations of the different values of the random variable from the expected value

The variance of a discrete random variable is:

xAll

XX xpx 22

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Standard Deviation

The standard deviation is the positive square root of the variance

The variance and standard deviation measure the spread of the values of the random variable from their expected value

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Example

Table 5.2

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Example Continued

89.

02.01.2505.01.2420.01.23

50.01.2220.01.2103.01.20

1.2

02.505.420.350.220.103.0

222

222

22

xAll xx

xAllx

xpx

xxp

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5.3 The Binomial Distribution

The binomial experiment…1. Experiment consists of n identical trials2. Each trial results in either “success” or

“failure”3. Probability of success, p, is constant from

trial to trial4. Trials are independent

If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable

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Binomial Distribution Continued

x-nxqp

x-nx

n =xp

!!

!

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Example 5.9

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x 0.05 0.1 0.15 … 0.500 0.8145 0.6561 0.5220 … 0.0625 41 0.1715 0.2916 0.3685 … 0.2500 32 0.0135 0.0486 0.0975 … 0.3750 23 0.0005 0.0036 0.0115 … 0.2500 14 0.0000 0.0001 0.0005 … 0.0625 0

0.95 0.9 0.85 … 0.50 x

Binomial Probability Table

values of p (.05 to .50)

values of p (.05 to .50)

P(x = 2) = 0.0486

p = 0.1

Table 5.7 (a)

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Several Binomial Distributions

Figure 5.6

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Mean and Variance of a Binomial Random Variable

npq

npq

np

xX

x

x

2

2

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Example

6164.38.

38.05.95.8

6.795.8

2

2

xX

x

x

npq

np

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5.4 The Poisson Distribution (Optional)

Consider the number of times an event occurs over an interval of time or space, and assume that

1. The probability of occurrence is the same for any intervals of equal length

2. The occurrence in any interval is independent of an occurrence in any non-overlapping interval

If x = the number of occurrences in a specified interval, then x is a Poisson random variable

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The Poisson Distribution Continued

!x

exp

x

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Poisson Probability Table

Table 5.9

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Poisson Probability Calculations

Table 5.10

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Mean and Variance of a Poisson Random Variable

Mean x = Variance 2

x = Standard deviation x is square

root of variance 2x

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Several Poisson Distributions

Figure 5.9