Page 92 - Probability, Random Variables and Random Processes

84 MULTIPLE RANDOM VARIABLES [CHAP 3

As in the discrete case, if X and Y are independent, then by Eq. (3.32),
~YIX(Y I X) = fY(Y) and fxlY(x I Y) = fx(x)

3.7 COVARIANCE AND CORRELATION COEFFICIENT
The (k, n)th moment of a bivariate r.v. (X, Y) is defined by

(discrete case)
mkn = E(xkYn) = (3.43)
xkyn fxy(x, y) dx dy (continuous case)

If n = 0, we obtain the kth moment of X, and if k = 0, we obtain the nth moment of Y. Thus,
m,, = E(X) = px and m,, = E(Y) = py (3.44)

If (X, Y) is a discrete bivariate r.v., then using Eqs. (3.43), (3.20), and (3.21), we. obtain

Similarly, we have
E(x2) = 1 1xi2~xAxi 9 Yj) = 1 xi2PAxi)
Yj Xi Xi
E(Y2) = z 1 Y~PXAX~ Yj) = C Y?P~Y~)
9
Yj Xi Yj
If (X, Y) is a continuous bivariate rev., then using Eqs. (3.43), (3.30), and (3.31), we obtain

Similarly, we have