Page 117 - Probability, Random Variables and Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

which results in the inequality

3.36. Verify Eq. (3.54).
From the Cauchy-Schwarz inequality [Eq. (3.97)], we have
{ECG - PX)(Y - PY)I)~ 5 EC(X - PX)~IEC(Y - PY)~I
or oxy2 I ox20y2

Then

Since pxy is a real number, this implies
p Y or -lIpxYIl

3.37. Let (X, Y) be the bivariate r.v. of Prob. 3.12.

(a) Find the mean and the variance of X.
(b) Find the mean and the variance of Y.
(c) Find the covariance of X and Y.
(d) Find the correlation coefficient of X and Y.
(a) From the results of Prob. 3.12, the mean and the variance of X are evaluated as follows:

(b) Similarly, the mean and the variance of Y are

By Eq. (3.51), the covariance of X and Y is

(d) By Eq. (3.53), the correlation coefficient of X and Y is

3.38. Suppose that a bivariate r.v. (X, Y) is uniformly distributed over a unit circle (Prob. 3.21).