Page 148 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 141

Thus, fz(z)={;'"' ""'I
otherwise

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

Thus, by Eq. (4.23), we obtain

.fzw(z, w) = I - I fxy(z-9 w)
and the marginal pdf of Z is

4.24. Let X and Y be independent standard normal r.v.'s. Find the pdf of Z = X/Y.
Since X and Y are independent, using Eq. (4.84), we have

which is the pdf of a Cauchy r.v. with parameter 1.

X and Y be two r.v.'s with joint pdf fxy(x, y) and joint cdf FXy(x, y). Let Z = max(X, Y).

Find the cdf of 2.
Find the pdf of Z if X and Y are independent.

The region in the xy plane corresponding to the event (max(X, Y) 5 zj is shown as the shaded area in
Fig. 4-8. Then
F,(z) = P(Z I Z) = P(X 5 Z, Y 5 Z) = FXy(z, Z) (4.85)
If X and Y are independent, then

F&) = Fx(z)Fy(z)
and differentiating with respect to z gives
fz(4 = fx(z)Fr(z) + Fx(z)fr(z)