Page 90 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 33 MULTIPLE RANDOM VARIABLES

D. Independent Random Variables :
If X and Y are independent r.v.'s, by Eq. (3.4),
Fxy(x, Y) = Fx(x)Fy(y)

Then

analogous with Eq. (3.22) for the discrete case. Thus, we say that the continuous r.v.'s X and Y are
independent r.v.'s if and only if Eq. (3.32) is satisfied.

3.6 CONDITIONAL DISTRIBUTIONS
A. Conditional Probability Mass Functions:
If (X, Y) is a discrete bivariate r.v. with joint pmf pxdxi, yj), then the conditional pmf of Y, given
that X = xi, is defined by

Similarly, we can define pxly(xi I yj) as

B. Properties ofpYlhj [xi):

1
1. 0 I pYlx(yj xi) 5 1
2. PY~X(Y~ = 1
l
xi)
yi
Notice that if X and Y are independent, then by Eq. (3.22),

C. Conditional Probability Density Functions:
If (X, Y) is a continuous bvivariate r.v. with joint pdf fxy(x, y), then the conditional pdf of Y, given
that X = x, is defined by

Similarly, we can define fxly(x I y) as

D. Properties of fy , &J 1 x):