Page 126 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

3.57. Let the joint pmf of (X, Y) be given by
xi = 1, 2, 3; yj = 1, 2
+
Pxy(xi, Yj) = {ri ~j)
otherwise
where k is a constant.
(a) Find the value of k.
(b) Find the marginal pmf's of X and Y.
Ans. (a) k = &
(b) px(xi) = &2xi + 3) xi = 1, 2, 3

3.58. The joint pdf of (X, Y) is given by
x > 0, y > 0
otherwise
where k is a constant.
(a) Find the value of k.
(b) Find P(X > 1, Y < I), P(X < Y), and P(X s 2).
Ans. (a) k = 2
(b) P(X > 1, Y < 1) = e-' - e-3 z 0.318; P(X < Y) = 4; P(X I 2) = 1 - e-2 z 0.865

3.59. Let (X, Y) be a bivariate r.v., where X is a uniform r.v. over (0, 0.2) and Y is an exponential r.v. with
parameter 5, and X and Y are independent.
(a) Find the joint pdf of (X, Y).
(b) Find P(Y 5 X).
0 < x < 0.2, y > 0
Ans. (a) fx&, y) =
otherwise

3.60. Let the joint pdf of (X, Y) be given by

otherwise
(a) Show that fxdx, y) satisfies Eq. (3.26).
(b) Find the marginal pdf's of X and Y.
Ans. (b) fx(x) = e-" x>O
1
fr(y) = 0' Y>O

3.61. The joint pdf of (X, Y) is given by
x<y<2x,o<x<2
otherwise
where k is a constant.
(a) Find the value of k.
(b) Find the marginal pdf's of X and Y.
Ans. (a) k =