Page 93 - Probability, Random Variables and Random Processes

CHAP. 3) MULTIPLE RANDOM VARIABLES

The variances of X and Y are easily obtained by using Eq. (2.31). The (1, 1)th joint moment of (X, Y),
m,, = E(XY) (3.49)
is called the correlation of X and Y. If E(XY) = 0, then we say that X and Y are orthogonal. The
covariance of X and Y, denoted by Cov(X, Y) or ax,, is defined by

Expanding Eq. (3.50), we obtain

If Cov(X, Y) = 0, then we say that X and Y are uncorrelated. From Eq. (3.51), we see that X and Y
are uncorrelated if

Note that if X and Y are independent, then it can be shown that they are uncorrelated (Prob.
3.32), but the converse is not true in general; that is, the fact that X and Y are uncorrelated does not,
in general, imply that they are independent (Probs. 3.33, 3.34, and 3.38). The correlation coefficient,
denoted by p(X, Y) or pxy, is defined by

It can be shown that (Prob. 3.36)

Note that the correlation coefficient of X and Y is a measure of linear dependence between X and Y
(see Prob. 4.40).

3.8 CONDITIONAL MEANS AND CONDITIONAL VARIANCES
If (X, Y) is a discrete bivariate r.v. with joint pmf pxy(xi, y,), then the conditional mean (or condi-
tional expectation) of Y, given that X = xi, is defined by

The conditional variance of Y, given that X = xi, is defined by

Yi
which can be reduced to

The conditional mean of X, given that Y = y,, and the conditional variance of X, given that Y = y,,
are given by similar expressions. Note that the conditional mean of Y, given that X = xi, is a func-
tion of xi alone. Similarly, the conditional mean of X, given that Y = yj, is a function of yj alone.
If (X, Y) is a continuous bivariate rev. with joint pdf fxy(x, y), the conditional mean of Y, given
that X = x, is defined by

The conditional variance of Y, given that X = x, is defined by