Page 145 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

The pdf s of X and Y are

Then, by Eq. (4.80a), we have

Now, z2 - 2zx + 2x2 = (ax - z/\/I)~ + z2/2, and we have

with the change of variables u = fix - z/& Since the integrand is the pdf of N(0; I), the integral is
equal to unity, and we get
1 e-zW - 1
-
fi(4 = - -z3/2(&2
&& &d
which is the pdf of N(0; 2). Thus, Z is a normal r.v. with zero mean and variance 2.

4.19. Let X and Y be independent uniform r.v.'s over (0, 1). Find and sketch the pdf of Z = X + Y.
Since X and Y are independent, we have

The range of Z is (0, 2), and
Fz(z) = P(X + Y I z) =
x+ysz
If 0 < z < 1 [Fig. 4-7(a)],
z
Fdz) = dx dy = shaded area = -
2
d
and fz(4 = , z
FZ(4
=
If 1 < z < 2 [Fig. 4-7(b)],
(2 - z)~
Fz(z) = 55 dx dy = shaded area = 1 - -
2

and

O<z<l
Hence,
otherwise