Page 133 - Probability, Random Variables and Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 125

where

which is the jacobian of the transformation (4.29).

4.5 EXPECTATION

A. Expectation of a Function of One Random Variable:
The expectation of Y = g(X) is given by

(1 g(x) fX(x) dx (continuous case)
-00

B. Expectation of a Function of More than One Random Variable:
Let XI, ..., Xn be n r.v.'s, and let Y = g(X,, ..., Xn).Then
(discrete case)

(continuous case)
(4.34)

C. Linearity Property of Expectation:
Note that the expectation operation is linear (Prob. 4.39), and we have

where a,'s are constants. If r.v.'s X and Y are independent, then we have (Prob. 4.41)

ECg(X)h( Y)I = ECg(X)IE[h( Y)I (4.36)
The relation (4.36) can be generalized to a mutually independent set of n r.v.'s XI, . . . , X,:

~[fi = i= fi ECg/Xi)I (4.37)
gi(xi)]
1