Page 212 - Probability, Random Variables and Random Processes

RANDOM PROCESSES [CHAP 5

WIENER PROCESSES

5.59. Let X,, . . . , X, be jointly normal r.v.'s. Show that the joint characteristic function of XI, . . . , X,
is given by
1
Y,, ... xn(ol, . . . , on) = exp wi pi - - oi a, o,
i= 1 2 i=l k=l
where pi = E(Xi) and aik = Cov(Xi, X,).
Let Y =alX, + a2X2 +-a. + anXn
By definition (4.50), the characteristic function of Y is

Now, by the results of Prob. 4.55, we see that Y is a normal r.v. with mean and variance given by [Eqs.
(4.108) and (4.1 1 I)]

Thus, by Eq. (4.125),

Equating Eqs. (5.176) and (5.1 73) and setting o = 1, we get

By replacing a,'s with mi's, we obtain Eq. (5.1 72); that is,
1" "
YXI ... X,(ol, . . . , an) = exp mi pi - - C C ai cok cik
i= 1 2 i=l ,'=I

Let

Then we can write

and Eq. (5.1 72) can be expressed more compactly as

5.60. Let XI, . . . , X, be jointly normal r.v.'s Let

where aik (i = 1, . . ., m; j = 1, . . . , n) are constants. Show that Y,, . . ., Y, are also jointly normal
r.v.'s.