Page 204 - Introduction to Statistical Pattern Recognition

186 Introduction to Statistical Pattern Recognition

I
E{M,.--M,.)(M,.-M,.)'} -A
=
N'
or
xp
E { Amy)Am(!) ] = -&j (5.13)
J
N
where hj'" is the ith diagonal component of A,..

(4) The third order central moments of a normal distribution are zero:
E { Am?Acf;) } = 0 . (5.14)

(5) The fourth order central moments of a normal distribution are [see
(2.57), (2.59), and (2.60)]:

0 otherwise .
Note that, in the equal index case of (5.15), N-1 is replaced by N for simpli-
city.

Moments off: Although we have not shown the higher order moments
of yi's other than the second, it is not so difficult to generalize the discussion
to obtain

(5.16)
i=l
and
E{O"'} =E{0(3'} = . . . =o,

where 0") is the ith order term of the Taylor expansion in (5.1) [see Problem