Page 107 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

0
Fig. 3-6

Thus k = 8.
(b) By Eq. (3.30), the marginal pdf of X is

(0 otherwise
By Eq. (3.31), the marginal pdf of Y is

(0 otherwise
Since fx&, y) # fx(x) fy(y), X and Y are not independent.
Note that if the range space Rxy depends functionally on x or y, then X and Y cannot be indepen-
dent r.v.'s.

3.20. The joint pdf of a bivariate r.v. (X, Y) is given by
k O<y<x<l
0 otherwise
where k is a constant.
(a) Determine the value of k.
(b) Find the marginal pdf's of X and Y.
(c) Find P(0 < X < i,O c Y < i).
(a) The range space Rxy is shown in Fig. 3-7. By Eq. (3.26),

Thus k = 2.
(b) By Eq. (3.30), the marginal pdf of X is

(0 otherwise
By Eq. (3.31)' the marginal pdf of Y is

(0 otherwise