Page 116 - Probability, Random Variables and Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

Thus, X and Y are uncorrelated.

3.34. Let (X, Y) be a bivariate r.v. with the joint pdf

Show that X and Y are not independent but are uncorrelated.
By Eq. (3.30), the marginal pdf of X is

Noting that the integrand of the first integral in the above expression is the pdf of N(0; 1) and the second
integral in the above expression is the variance of N(0; I), we have

Since fx,(x, y) is symmetric in x and y, we have
I

Now fx,(x, y) # fx(x) fu(y), and hence X and Y are not independent. Next, by Eqs. (3.47a) and (3.473),

since for each integral the integrand is an odd function. By Eq. (3.43),

The integral vanishes because the contributions of the second and the fourth quadrants cancel those of the
first and the third. Thus, E(XY) = E(X)E(Y), and so X and Y are uncorrelated.

3.35. Let (X, Y) be a bivariate r.v. Show that

[E(xy)12 5 E(x2)E(y2)
This is known as the Cauchy-Schwarz inequality.
Consider the expression E[(X - for any two r.v.'s X and Y and a real variable a. This expres-
sion, when viewed as a quadratic in a, is greater than or equal to zero; that is,
E [(X - a Y)2] 2 0
for any value of a. Expanding this, we obtain
E(X2) - 2aE(XY) + a2E(Y2) 2 0
Choose a value of a for which the left-hand side of this inequality is minimum,