Page 121 - Schaum's Outlines - Probability, Random Variables And Random Processes

MULTIPLE RANDOM VARIABLES [CHAP 3

ps) where pi = 4 (i = 1,2, . . . ,6). Hence, by Eq. (3.87),

3.48. Show that the pmf of a multinomial r.v. given by Eq. (3.87) satisfies the condition (3.68); that is,
-
= l
,
. 1 Pxlx2 -. ~ ( X, ~ . -7 ~ k ) (3.1 01)
~
1
where the summation is over the set of all nonnegative integers x,, x, , . . . , xk whose sum is n.
The multinomial theorem (which is an extension of the binomial theorem) states that

where x, + x, + . . . + x, = n and

is called the multinomial coefficient, and the summation is over the set of all nonnegative integers x,, x, , . . . ,
x, whose sum is n.
Thus, setting ai = pi in Eq. (3.102), we obtain

3.49. Let (X, Y) be a bivariate normal r.v. with its pdf given by Eq. (3.88).
(a) Find the marginal pdf's of X and Y.
(b) Show that X and Y are independent when p = 0.
(a) By Eq. (3.30), the marginal pdf of X is

From Eqs. (3.88) and (3.89), we have

Rewriting q(x, y),

1 1
Then f~x) exp[ 5 q,(x, i)] d~
=

where