Page 139 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

Fig. 4-3

The range Ry is found as follows: From Fig. 4-3, we see that
For a > 0: R,={y:b<y<a+b)
For a < 0: R,={y:a+b<y<b)

4.5. Let Y = ax + b. Show that if X = N(p; a2), then Y = N(ap + b; a202), and find the values of a
and b so that Y = N(0; 1).
Since X = N(p; a2), by Eq. (2.52),
1 1
fX(4 = ---
,pi.
Hence, by Eq. (4.70),

which is the pdf of N(ap + b; a2a2). Hence, Y = N(ap + b; a2a2). Next, let ap + b = 0 and a2a2 = 1, from
which we get a = 110 and b = - pla. Thus, Y = (X - p)/a is N(0; 1) (see Prob. 4.1).

4.6. Let X be a r.v. with pdf f,(x). Let Y = x2. Find the pdf of Y.
The event A = (Y 5 y) in R, is equivalent to the event B = (-& 5 X 5 &) in Rx (Fig. 4-4). If
y 1 0, then

and

Thus,