Page 87 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 69

~=E{z:) -E2{z?] =E(z:] - 1. (3.55)

For normal distributions, when the zits are uncorrelated, they are also indepen-
dent. Therefore, (3.55) can be used to compute Var{d2}, and y= 2. Figure
3-8 shows the distribution of d2 with the mean n and the standard deviation
G.

Example 6: Let the xi's be mutually independent and identically distri-
buted with a gamma densify function, which is characterized by two parameters
a and p as in (2.54). Using m = E { xi } and 02 = Var{ xi 1, (3.51) becomes

(3.56)

Then, y is
E (xi-m14 - d4
Y=
CY4
6
=2+- (3.57)
p+l'
where the second line is obtained by using the mth order moments of a gamma
density as

(3.58)

An exponential distribution is a special case of a gamma distribution with
p = 0, for which y becomes 8. On the other hand, y = 2 is obtained by letting
p be 00. Recall from (3.55) that y for a normal distribution is 2.

Example 7: In (3.52) and (3.54), only the first and second order
moments of d2 are given. However, if the 2;'s are normal, the density function
of d2 is known as [7].

(3.59)

which is the gamma density with p = n /2 - 1 and a = 1/2.