Page 106 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

Fig. 3-5
(a) The range space Rxy is shown in Fig. 3-5(a). By Eq. (3.26),

Thus k = 4.
(b) To determine whether X and Y are independent, we must find the marginal pdf's of X and Y. By Eq.
(3.30),

(0 otherwise
By symmetry,
fdy) = {? O<y<l
otherwise
Since f,,(x, y) = fx(x) fJy), X and Y are independent.
(c) The region in the xy plane corresponding to the event (X +. Y < 1) is shown in Fig. 3-5(b) as a shaded
area. Then
x2 1-Y
dx dY = [4Y(T lo ) dY
= l4YMl - Y)'] dy = t

3.19. The joint pdf of a bivariate r.v. (X, Y) is given by
O<x<y<l
otherwise

where k is a constant.
(a) Find the value of k.
(b) Are X and Y independent?

(a) The range space Rxy shown in Fig. 3-6. By Eq. (3.26),
is