Page 114 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

COVARIANCE AND CORRELATION COEFFICIENTS
3.32. Let (X, Y) be a bivariate r.v. If X and Y are independent, show that X and Y are uncorrelated.

If (X, Y) is a discrete bivariate r.v., then by Eqs. (3.43) and (3.22)'

If (X, Y) is a continuous bivariate r.v., then by Eqs. (3.43) and (3.32),

Thus, X and Y are uncorrelated by Eq. (3.52).

3.33. Suppose the joint pmf of a bivariate r.v. (X, Y) is given by

(a) Are X and Y independent?
(b) Are X and Y uncorrelated?
(a) By Eq. (3.20), the marginal pmf's of X are

By Eq. (3.21), the marginal pmf's of Y are

Thus X and Y are not independent.
(b) By Eq.s (3.45a), (3.45b), and (3.43), we have

Now by Eq. (3.51),