Page 31 - Introduction to Statistical Pattern Recognition

2 Random Vectors and their Properties 13

This relation between qi(X) and pj(X) provides a basic tool in hypothesis test-
ing which will be discussed in Chapter 3.

Parameters of Distributions

A random vector X is fully characterized by its distribution or density
function. Often, however, these functions cannot be easily determined or they
are mathematically too complex to be of practical use. Therefore, it is some-
times preferable to adopt a less complete, but more computable, characteriza-
tion.

Expected vector: One of the most important parameters is the expected
wctor’ or mean of a random vector X. The expected vector of a random vector
X is defined by

M = E{XJ =JXp(X) dX , (2.9)
where the integration is taken over the entire X-space unless otherwise
specified.
The ith component of M, m,, can be calculated by
+m
rn, = Jx,p(~) dx = j x,p(s,) dx, , (2.10)
-m
is
where p (s,) the marginal density of the ith component of X, given by
.
p (SI) = I” . . j+-p (X) dx I . . . d,Vl -, dx, +, . . . dx,, . (2.1 1)
-_ -m
I1 - I
Thus, each component of M is actually calculated as the expected value of an
individual variable with the marginal one-dimensional density.

The conditional expected \vector of a random vector X for 0, is the
integral

MI =/?{XI 0,) =JXp,(X)dX, (2.12)
where p,(X) is used instead of p (X) in (2.9).

Covariance matrix: Another important set of parameters is that which
indicates the dispersion of the distribution. The coiwriance mafrk of X is