Page 123 - Probability, Random Variables and Random Processes

CHAP. 3) MULTIPLE RANDOM VARIABLES 115

Comparing the integrand with Eq. (2.52), we see that the integrand is a normal pdf with mean
pY + p(oy/oX)(x - pX) and variance (1 - p2)oy2. Thus, the integral must be unity and we obtain

In a similar manner, the marginal pdf of Y is

(b) When p = 0, Eq. (3.88) reduces to

Hence, X and Y are independent.

3.50. Show that p in Eq. (3.88) is the correlation coefficient of X and Y.
By Eqs. (3.50) and (3.53), the correlation coefficient of X and Y is

where fxy(x, y) is given by Eq. (3.88). By making a change in variables v = (x - px)/ax and w = (y - py)/oy,
we can write Eq. (3.105) as
I
1 1
-
(v2
PXY = J:' j;/w 2,1 - p2)i/2 exp - - 2pvw + w2) do dw
[ 2u - p2)

The term in the curly braces is identified as the mean of V = N(pw; 1 - p2), and so

The last integral is the variance of W = N(0; l), and so it is equal to 1 and we obtain pxy = p.

3.51. Let (X, Y) be a bivariate normal r.v. with its pdf given by Eq. (3.88). Determine E(Y I x).
By Eq. (3.58),

where

Substituting Eqs. (3.88) and (3.103) into Eq. (3.107), and after somecancellation and rearranging, we obtain