Page 141 - Probability, Random Variables and Random Processes

CHAP. 4) FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 133

Fig. 4-4
Alternative Solution :
If y c 0, then the equation y = x2 has no real solutions; hence f,(y) = 0. If y > 0, then y = x2 has two
solutions, x, = fi and x, = -A. Now, y = g(x) = x2 and gf(x) = 2x. Hence, by Eq. (4.8),

4.7. Let Y = x2. Find the pdf of Y if X = N(0; 1).
Since X = N(0; 1)

Since f,(x) is an even function, by Eq. (4.74), we obtain

4.8. Let Y = X2. Find and sketch the pdf of Y if X is a unifonn rev. over (- 1, 2).
The pdf of X is [Eq. (2.4411 [Fig. 4-5(a)]
= {a otherwise
-l<x<2

In this case, the range of Y is (0, 4), and we must be careful in applying Eq. (4.74). When 0 < y c 1, both
& and -& are in Rx = (- 1, 2), and by Eq. (4.74),

When I < I. c 4, & is in R, = (- 1, 2) but -A < - 1, and by Eq. (4.74),

O<y<l
3J;
Thus,
l<y<4
otherwise
which is sketched in Fig. 4-5(b).