Page 105 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

Thus, ~(~+y<4)=1-~(~+~>4)=1-&=~

CONTINUOUS BIVARIATE RANDOM VARIABLES-PROBABILITY DENSITY
FUNCTIONS
3.17. The joint pdf of a bivariate r.v. (X, Y) is given by
o<x<2,o<y<2
= g(.
otherwise
,Ax. , +
where k is a constant.
(a) Find the value of k.
(b) Find the marginal pdf's of X and Y.
(c) Are X and Y independent?

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

O<x<2
Y = o otherwise
Since fxy(x, y) is symmetric with respect to x and y, the marginal pdf of Y is
MY)={;(~+l) O<Y<2
otherwise
(c) Since fx,(x, y) # fx(x) fy(y), X and Y are not independent.

3.18. The joint pdf of a bivariate r.v. (X, Y) is given by
O<x<l,O<y<l
otherwise
where k is a constant.
(a) Find the value of k.
(b) Are X and Y independent?
(c) Find P(X + Y < 1).