Page 139 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 121

(d) Plot the error-reject curve.
5. Two normal distributions are characterized by
PI = 0.6, P2 = 0.4 ,

ComputetheBayeserrorforcll =c22 =OandcI2 =c2].
6. Show how to derive the variances of (3.144) and (3.145) for normal dis-
tributions.
7. Let xi (i=l, . . . ,n) be independent and identically distributed random
variables, whose distributions are exponential with the parameters a, for
o1 and a2 for w;?. Find E [ h (X) wj ) where h (X) is the quadratic equa-
I
tion of (3.1 l).
8. The equivocation is given by

Prove that the equivocation is larger than the asymptotic nearest neighbor
error but smaller than the Bhattacharyya error bound.
9. When two distributions are normal with an equal covariance matrix, Z,
both the Bayes error, E, and the Bhattacharyya bound, E,, are expressed
as functions of 1 = (M2-M l)TZ-' (M2-M). Plot E and E, vs. 1.
10. Three distributions are normal with

The cost matrix is