Page 124 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 31 MULTIPLE RANDOM VARIABLES

Note that the left-hand side of Eq. (3.11 1) depends only on x, whereas the right-hand side depends only on
x2 + y2; thus

where c is a constant. Rewriting Eq. (3.1 12) as

and integrating both sides, we get

where a and k are constants. By the properties of a pdf, the constant c must be negative, and setting c =
- l/a2, we have

Thus, by Eq. (2.52), X = N(0; a2) and

,-~2n202)
fX(4 = -
fia
In a similar way, we can obtain the pdf of Y as

Since X and Y are independent, the joint pdf of (X, Y) is
1
fXy(x, y) = fX(x) fy(y) = 7 e-(x2+ y2)1(2a2)
2aa
which indicates that (X, Y) is a bivariate normal r.v.

3.54. Let (XI, X,, .. ., X,) be an n-variate normal r.v. with its joint pdf given by Eq. (3.92). Show that
if the covariance of Xi and Xj is zero for i # j, that is,

then XI, X, , . . . , X, are independent.
From Eq. (3.94) with Eq. (3.1 14), the covariance matrix K becomes

It therefore follows that

and