Page 113 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

Again note that fyldy I x) = fy(y) and fxldx I y) = fx(x), as must be the case since X and Y are independent,
as shown in Prob. 3.18.

3.30. Find the conditional pdf's fylx(y I x) and fxly(x I y) for the bivariate r.v. (X, Y) of Prob. 3.20.
From the results of Prob. 3.20, we have
2 O<ylx<l
fxy(x' = i 0 otherwise
fx(x) = 2x O<x<l
fY(Y) = 2(1 - Y) 0 < Y < 1
Thus, by Eqs. (3.38) and (3.39),
1
fr,x(Y I x) = ; y<x<l,O<x<l

3.31. The joint pdf of a bivariate r.v. (X, Y) is given by

x>O,y>O

otherwise
(a) Show that f,,(x, y) satisfies Eq. (3.26).
(b) Find P(X > 1 I Y = y).
(a) We have

(b) First we must find the marginal pdf on Y. By Eq. (3.31),

By Eq. (3.39), the conditional pdf of X is

x>o,y>o
fxlu(x I Y) = - -
otherwise
Then