Page 41 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 23

and

= 0 except (i=k and j=t) . (2.52)

The reader may confirm (2.52) for all possible combinations of i, j, k, and 1. When
11,. A
i=kandj=t,Cov(cjj, ck.:) becomesVar(cjj), whichisgiven in (2.50).
L) A
Also, the covariance between mi and ckt may be computed in approximation
as follows:
1
Cov(m,,ck, E - Cov{x,, xkx, 1
A11
1
N
= --[E{X,X,R) - E(Xj)E(XkX t1
1
N
=0, (2.53)

because E ( xjxkx, = 0 and E { xi } = 0 for a zero-mean normal distribution.
}

Normal case without approximation: When samples are drawn from a
L)
normal distribution, the variances and covariances for cjj of (2.45) are known
without approximation. In order to see the effects of the approximation on (2.50)-
(2.52), let us study the exact solutions here. Again, for simplicity, let us assume a
zero expected vector and a diagonal covariance matrix A.
It is known that the sample variance ijj = I/(N-l)E:=, (xi, -mi)* for a nor-
mal xi has a gamma distribution as [2]

(2.54)

where
N-1
p+l=- N-' and ai=- (2.55)
2 2hi '
and r(.) the gamma function and u (-) is a step function. The expected value and
is
variance of (2.54) are also known as