Fundamentals of Probability and Statistics for Engineers

5 FUNCTIONS OF RANDOM VARIABLES

The basic topic to be discussed in this chapter is one of determining the relation-

ship between probability distributions of two random variables X and Y when

they are related by Y g(X). The functional form of g(X) is given and determin-

istic. Generalizing to the case of many random variables, we are interested in the

determination of the joint probability distribution of Yj, j 1, 2,... , m, which is

functionally dependent on Xk, k 1, 2,... , n, according to

when the joint probabilistic behavior of Xk,k 1,2,...,n, is known.

Some problems of this type (i.e. transformations of random variables) have

been addressed in several places in Chapter 4. For example, Example 4.11 con-

siders transformation Y X 1 Xn, and Example 4.18 deals with the trans-

formation of 3 n random variables (X 1 ,X 2 ,...,X 3 n)totworandomvariables

(X^0 ,Y^0 ) defined by Equations (4.90). In science and engineering, most phenomena

are based on functional relationships in which one or more dependent variables

are expressed in terms of one or more independent variables. For example, force is

a function of cross-sectional area and stress, distance traveled over a time interval

is a function of the velocity, and so on. The techniques presented in this chapter

thus permit us to determine the probabilistic beha vior of random variables that

are functionally dependent on some others with known probabilistic properties.

In what follows, transformations of random variables are treated in a systemat-

ic manner. In Equation (5.1), we are basically interested in the joint distributions

and joint moments of Y 1 ,...,Ym given appropriate information on X 1 ,...,Xn.

5.1 Functions of One Random Variable

Consider first a simple transformation involving only one random variable, and let

Fundamentals of Probability and Statistics for Engineers (^) T.T. Soong 2004 John Wiley & Sons, Ltd
ISBN s: 0-470-86813-9 (H B) 0-470-86814-7 (PB)

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Yjˆgj…X 1 ;...;Xn†;jˆ 1 ; 2 ;...;m;mn;… 5 : 1 †

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Yˆg…X†… 5 : 2 †