Page 128 - Probability, Random Variables and Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

(b) fx(x) = &x3(4 - $x) O<x<2
O<y<2
Qy3) 2 < Y < 4
otherwise
3.62. The joint pdf of (X, Y) is given by

(a) Find the marginal pdf's of X and Y.
(b) Are X and Y independent?
Ans. (a) fx(x) = xe-"'I2 x > 0
fky) = ye-y212 y > 0
(b) Yes

3.63. The joint pdf of (X, Y) is given by
x>O, y>o
fXYk Y) =
otherwise
(a) Are X and Y independent?
(b) Find the conditional pdf's of X and Y.
Ans. (a) Yes

3.64. The joint pdf of (X, Y) is given by

otherwise
(a) Find the conditional pdf's of Y, given that X = x.
(b) Find the conditional cdf's of Y, given that X = x.
1
Ans. (a) fylx(y x) = ex-Y y2x

3.65. Consider the bivariate r.v. (X, Y) of Prob. 3.14.
(a) Find the mean and the variance of X.
(b) Find the mean and the variance of Y.
(c) Find the covariance of X and Y.
(d) Find the correlation coefficient of X and Y.
Ans. (a) E(X) = g, Var(X) =
(b) E(Y)= y,Var(Y)= 3
(c) Cov(X, Y) = -A
(d) p = -0.025
3.66. Consider a bivariate r.v. (X, Y) with joint pdf

Find P[(X, Y) I x2 + y2 I a2].