Page 284 - Probability, Random Variables and Random Processes

DECISION THEORY [CHAP. 8

Then, by setting Coo = C,, = 0, C,, = 2, and C,, = 1 in Eq. (8.19), the minimum Bayes' risk C* can be
expressed as a function of P(Ho) as

C*[P(Ho)] = P(Ho) I:4rxl dx + Z[1 - P(Ho)] [(:e2. dx + [me-2x dx]

e-'. dx
= P(Ho) [e-. dx + 1[1 - P(Ho)] i'
= P(Ho)(l - e-*) + 2[1 - P(Ho)]e-26
From the definition of 6 [Eq. (8.34)], we have

Thus e-d = PWo) and e-2*= p2(Ho)
4[1 - P(H0)I 16[1 - P(HO)l2
Substituting these values in to Eq. (8.35), we obtain

Now the value of P(Ho) which maximizes C* can be obtained by setting dC*[P(Ho)J/dP(Ho) equal to zero
and solving for P(Ho). The result yields P(Ho) = 3. Substituting this value into Eq. (8.34), we obtain the
following minimax test :

8.13. Suppose that we have n observations Xi, i = 1, . . ., n, of radar signals, and Xi are normal iid
r.v.'s under each hypothesis. Under H,, Xi have mean p, and variance a2, while under HI, Xi
have mean p, and variance a2, and p, > p, . Determine the maximum likelihood test.
By Eq. (2.52) for each Xi, we have

Since the Xi are independent, we have

With, ie likelihood ratio is then given by

Hence, the maximum likelihood test is given by