Page 282 - Probability, Random Variables and Random Processes

DECISION THEORY [CHAP. 8

By Eq. (8.19)' the Bayes' risk is

Now we can express

Then C can be expressed as

Since Ro u R, = S and Ro n R, = 4, we can write

Then Eq. (8.32) becomes

I
e = coo P(H0) + COIP(H1) + K(C10 - Coo)P(Ho)f (x I HOI1 - WOl - Cl,)P(Hl)f (x H,II) dx
JR
The only variable in the above expression is the critical region R,. By the assumptions [Eq. (8.20)] Clo >
Coo and Col > C,,, the two terms inside the brackets in the integral are both positive. Thus, C is mini-
mized if R, is chosen such that

for all x E R ,. That is, we decide to accept H, if

In terms of the likelihood ratio, we obtain

which is Eq. (8.21).

8.11. Consider a binary decision problem with the following conditional pdf's:
f(x I H,) = ie-IXI
f(x(H1) = e-21XI
The Bayes' costs are given by

Coo = Cll = 0 C,, = 2 Cl0 = 1
(a) Determine the Bayes' test if P(Ho) = 3 and the associated Bayes' risk.
(b) Repeat (a) with P(H,) = *.

(a) The likelihood ratio is

By Eq. (8.21)' the Bayes' test is given by