Page 96 - Probability, Random Variables and Random Processes

88 MULTIPLE RANDOM VARIABLES [CHAP 3

The covariance of Xi and Xj is defined as
oij = Cov(Xi, Xi) = E[(Xi - pi)(Xj - pj)]
The correlation coefficient of Xi and Xj is defined as
Cov(Xi,Xj) - 0..
V
Pij =
oi oj ai oj

3.10 SPECIAL DISTRIBUTIONS
A. Multinomial Distribution:
The multinomial distribution is an extension of the binomial distribution. An experiment is
termed a multinomial trial with parameters p,, p, , . . . , p,, if it has the following conditions:
1. The experiment has k possible outcomes that are mutually exclusive and exhaustive, say A,, A,,
..., Ak.
k
2. P(Ai) = pi i = 1, . . . , k and 1 pi = 1 (3.86)
i= 1
Consider an experiment which consists of n repeated, independent, multinomial trials with param-
eters p,, p,, . . . , p,. Let Xi be the r.v. denoting the number of trials which result in Ai. Then (X,, X, ,
. . . , X& is called the multinomial r.v. with parameters (n, p,, p, , . . . , pk) and its pmf is given by (Prob.
3.46)

k
for xi = 0, 1, . . . , n, i = 1, . . . , k, such that xi = n.
i= 1
Note that when k = 2, the multinomial distribution reduces to the binomial distribution.

B. Bivariate Normal Distribution:
A bivariate r.v. (X, Y) is said to be a bivariate normal (or gaussian) r.v. if its joint pdf is given by

[ (
where - 2P(3(7) + (yy]
(x, ) =
and A, pY, ox2, oy2 are the means and variances of X and Y, respectively. It can be shown that p is
the correlation coefficient of X and Y (Prob. 3.50) and that X and Y are independent when p = 0
(Prob. 3.49).

C. N-variate Normal Distribution:
Let (XI, . . . , X,) be an n-variate r.v. defined on a sample space S. Let X be an n-dimensional
random vector expressed as an n x 1 matrix: