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Chapter 07

School

York UniversityDepartment

Operations Management and Information SystemCourse Code

OMIS 2010Professor

Alan MarshallChapter

07This

**preview**shows pages 1-3. to view the full**21 pages of the document.**Chapter 7: Random Variables and Discrete Probability Distributions

7.1 Random Variables and Probability Distributions

This section introduced the concept of a random variable, which assigns a numerical value to each

simple event in a sample space, thereby enabling us to work with numerical-valued outcomes. A random

variable is discrete if the number of possible values that it can assume is finite or countably infinite;

otherwise, the random variable is continuous. The second important concept introduced was that of a

discrete probability distribution, which lists all possible values that a discrete random variable can take

on, together with the probability that each value will be assumed. When constructing a discrete probabil-

ity distribution, bear in mind that the probabilities must sum to 1.

Example 7.1

The local taxation office claims that 10% of the income tax forms processed this year contain errors.

Suppose that two of these forms are selected at random. Find the probability distribution of the random

variable X, defined as the number of forms selected that contain an error.

Solution

A sample space for this random experiment is

S =

{

}

21212121 ,,, FFFFFFFF

where, for example, 21FF indicates that the first form selected contains an error and the second form

does not. The random variable X assigns a value of 0, 1, or 2 to each of these simple events. To find the

probability that each of these values will be assumed, we must determine the simple events that result in

each of these values being assumed and the probabilities that the simple events will occur. For example,

X = 2 if and only if the event 21FF occurs. Since the occurrence of an error on one form is independent

of the occurrence of an error on the other form,

01.)1)(.1(.)()()()2()2( 2121

=

=

=

=== FPFPFFPpXP

Similarly,

81.)9)(.9(.)()()()0()0( 18.)1)(.9(.)9)(.1(. )()()()( )()()1()1(

2121

2121

2121

======

=+=

+=

+===

FPFPFFPpXP

FPFPFPFP

FFPFFPpXP

Summarizing the probability distribution of X in tabular form, we obtain

72

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x p(x)

0 .81

1 .18

1 .01

Note that we could have used a probability tree to identify the simple events and their associated

probabilities.

EXERCISES

7.1 For each of the following random variables, indicate whether the variable is discrete or continu-

ous and specify the possible values that it can assume.

a) X = the number of customers served by a restaurant on a given day

b) X = the time in minutes required to complete a particular assembly

c) X = the number of accidents at a particular intersection in a given week

d) X = the number of questions answered correctly by a randomly selected student who wrote

a quiz consisting of 15 multiple choice questions

e) X = the weight in ounces of a newborn baby chosen at random

7.2 The following table gives the probability distribution of X, the number of telephones in a ran-

domly selected home in a certain community.

x p(x)

0 .021

1 .412

2 .283

3 .188

73

d

c

d

d

c

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

4 .096

If one home is selected at random, find the probability that it will have:

a) no telephone;

b) fewer than two telephones;

c) at least three telephones;

d) between one and three telephones (inclusive).

7.3 The security force on a particular campus estimates that 80% of the cars that are parked illegally

on campus are detected and issued a $10 parking ticket. Suppose your car will be parked ille-

gally on two occasions during the coming year. Determine the probability distribution of the

random variable X = the total dollar amount of parking tickets that you will receive.

7.2 Expected Value and Variance

This section dealt with the two main descriptive measures of a discrete random variable: its expected

value and its variance. As well as being able to calculate the numerical values of these measures, you are

expected to have a general understanding of their meanings and to be familiar with the laws of expected

value and of variance.

74

0.21

0.433

0.188

0.979

X

p(X)

0,0

1,0

0,1

1,1

(.2)(.2)

(.8)(.2)

(.2)(.8)

(.8)(.8)

= 0.04

= 0.16

= 0.16

=0.64

= .32

X

p(x)

$0

$10

$20

0.04

0.32

0.64

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