Page 131 - Schaum's Outlines - Probability, Random Variables And Random Processes

124 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

then the joint pdf of Z and W is given by

fzw(z, W) = fXYk Y) I J(x, Y) I -
where x = q(z, w), y = r(z, w), and

which is the jacobian of the transformation (4.1 7). If we define

and Eq. (4.1 9) can be expressed as

4.4 FUNCTIONS OF n RANDOM VARIABLES
A. One Function of n Random Variables:
Given n r.v.'s X ,, . . . , X, and a function g(x,, . . . , x,), the expression

y = g(X1, ..., X,)
defines a new r.v. Y. Then

and
where Dy is the subset of the range of (X,, ..., X,) such that g(xl, ..., x,) < y. If XI, ..., X, are
continuous r.v.'s with joint pdf f,, ... ,AX . . . , xn), then

B. n Functions of n Random Variables:
When the joint pdf of n r.v.'s X,, . . . , X, is given and we want to determine the joint pdf of n r.v.'s
Y,, .. ., Y,, where
Yl = gl(X1, -7 Xn)
(4.28)
K = gn(X1, Xn)
the approach is the same as for two r.v.'s. We shall assume that the transformation