Page 136 - 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 ˆ 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 Y j , j ˆ 1, 2, . . . , m, which is
functionally dependent on X k , k ˆ 1, 2, . . . , n, according to

Y j ˆ g X 1 ; ... ; X n † ; j ˆ 1; 2; ... ; m; m n; 5:1†
j
when the joint probabilistic behavior of X k , k ˆ 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 ˆ X 1 ‡ ‡ X n , and Example 4.18 deals with the trans-
formation of 3n random variables (X 1 , X 2 , . . ., X 3n ) to two random variables
0
0
(X ,Y ) 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 behavior 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 , . . ., Y m given appropriate information on X 1 , . . ., X n .

5.1 FUNCTIONS OF ONE RANDOM VARIABLE

Consider first a simple transformation involving only one random variable, and let
Y ˆ g X† 5:2†

Fundamentalsof Probability and Statistics for Engineers T.T. Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)
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