Page 282 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 81 DECISION THEORY

Taking the natural logarithm of both sides of the last expression yields

Thus, the decision regions are given by

0.693 0.693
Then PI = P(Dl I Ho) =

P11 = P(Do 1 HI) = e2"dx+ e-"dx=2 ee-2"dx=0.25
1' 0.693 I' 0.693
and by Eq. (8.19), the Bayes' risk is

(b) The Bayes' test is

Again, taking the natural logarithm of both sides of the last expression yields

Thus, the decision regions are given by
Ro = {x: 1x1 > 1.386) R, = {x: 1x1 < 1.386)

Then

and by Eq. (8.19), the Bayes' risk is

8.12. Consider the binary decision problem of Prob. 8.11 with the same Bayes' costs. Determine the
minimax test.
From Eq. (8.33), the likelihood ratio is

In terms of P(Ho), the Bayes' test [Eq. (8.21)] becomes

Taking the natural logarithm of both sides of the last expression yields

For P(H,) > 0.8,6 becomes negative, and we always decide H,. For P(H,) _< 0.8, the decision regions are
Ro = (x: 1x1 > S) R, = {.x: 1x1 < 6)