Page 141 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

4.9. Let Y = ex. Find the pdf of Y if X is a uniform r.v. over (0, 1).
The pdf of X is
1 O<x<l
0 otherwise
The cdf of Y is

d d 1
Thus, fu(y)=-FAY)=-lny=- 1 < y<e
dy dy Y
Alternative Solution :
The function y = g(x) = ex is a continuous monotonically increasing function. Its inverse is
x = g-'Q = h(y) = In y. Thus, by Eq. (4.6), we obtain
t

(0 otherwise

4.10. Let Y = ex. Find the pdf of Y if X = N(p ; a2).
The pdf of X is [Eq. (2.52)]

Thus, using the technique shown in the alternative solution of Prob. 4.9, we obtain

Note that X = In Y is the normal r.v.; hence, the r.v. Y is called the log-normal r.v.