Page 147 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

where k is a constant which does not depend on z. The value of k is determined as follows: Using Eq. (2.22)
and definition (2.77) of the gamma function, we have

Hence, k = la+B/T(a + B) and

which indicates that Z is a gamma r.v. with parameters (a + /I, A).

4.21. Consider two r.v.'s X and Y with joint pdff,,(x, y). Determine the pdf of Z = XY.
Let Z = XY and W = X. The transformation z = xy, w = x has the inverse transformation x = w,
y = z/w, and

Thus, by Eq. (4.23), we obtain

and the marginal pdf of Z is

4.22. Let X and Y be independent uniform r.v.3 over (0, 1). Find the pdf of Z = XY.
We have
1 O<x<l,O<y<l
fx'x' = {O otherwise
The range of Z is (0, 1). Then
1
hr(l :) = { 0 O<w<1,0<z/w<1
otherwise
fxr(W3 3 = { 1 O<z<w<l
or
0 otherwise
By Eq. (4.82),