Applied Statistics and Probability for Engineers

142 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

CD MATERIAL

7. Determine the distribution of a function of one or more random variables


8. Calculate moment generating functions and use them to determine moments for random variables
   and use the uniqueness property to determine the distribution of a random variable


9. Provide bounds on probabilities for arbitrary distributions based on Chebyshev’s inequality

Answers for most odd numbered exercises are at the end of the book. Answers to exercises whose

numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete

worked solutions to certain exercises are also available in the e-Text. These are indicated in the

Answers to Selected Exercises section by a box around the exercise number. Exercises are also

available for the text sections that appear on CD only. These exercises may be found within the

e-Text immediately following the section they accompany.

In Chapters 3 and 4 we studied probability distributions for a single random variable. However,

it is often useful to have more than one random variable defined in a random experiment. For ex-

ample, in the classification of transmitted and received signals, each signal can be classified as

high, medium, or low quality. We might define the random variable Xto be the number of high-

quality signals received and the random variable Yto be the number of low-quality signals

received. In another example, the continuous random variable Xcan denote the length of one di-

mension of an injection-molded part, and the continuous random variable Ymight denote the

length of another dimension. We might be interested in probabilities that can be expressed in

terms of both Xand Y. For example, if the specifications for Xand Yare (2.95 to 3.05) and (7.60

to 7.80) millimeters, respectively, we might be interested in the probability that a part satisfies

both specifications; that is, P(2.95 X 3.05 and 7.60 Y 7.80).

In general, if Xand Yare two random variables, the probability distribution that defines

their simultaneous behavior is called a joint probability distribution.In this chapter, we

investigate some important properties of these joint distributions.

5-1 TWO DISCRETE RANDOM VARIABLES

5-1.1 Joint Probability Distributions

For simplicity, we begin by considering random experiments in which only two random vari-

ables are studied. In later sections, we generalize the presentation to the joint probability

distribution of more than two random variables.

EXAMPLE 5-1 In the development of a new receiver for the transmission of digital information, each re-

ceived bit is rated as acceptable, suspect,or unacceptable,depending on the quality of the

received signal, with probabilities 0.9, 0.08, and 0.02, respectively. Assume that the ratings of

each bit are independent.

In the first four bits transmitted, let

Xdenote the number of acceptable bits

Ydenote the number of suspect bits

Then, the distribution of Xis binomial with n4 and p0.9, and the distribution of Yis

binomial with n4 and p0.08. However, because only four bits are being rated, the possible

values of Xand Yare restricted to the points shown in the graph in Fig. 5-1. Although the possi-

ble values of Xare 0, 1, 2, 3, or 4, if y3, x0 or 1. By specifying the probability of each of

the points in Fig. 5-1, we specify the joint probability distribution of Xand Y. Similarly to an in-

dividual random variable, we define the range of the random variables (X, Y) to be the set of

points (x, y) in two-dimensional space for which the probability that Xxand Yyis positive.

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