Page 280 - Probability, Random Variables and Random Processes

DECISION THEORY [CHAP. 8

Then by Eqs. (8.1) and (8.2) and using Table A (Appendix A), we obtain

8.7. In the binary communication system of Prob. 8.6, suppose that P(H,) = and P(H,) = 4.
(a) Using the MAP test, determine which signal is transmitted when x = 0.6.
(b) Find PI and PI,.
(a) Using the result of Prob. 8.6 and Eq. (8.15), the MAP test is given by

Taking the natural logarithm of the above expression, we get

Since x = 0.6 < 1.193, we determine that signal s,(t) was transmitted.
(b) The decision regions are given by
Ro = {x: x < 1.193) = (-MI, 1.193)
R1 = {x: x > 1.193) = (1.193, a)

Thus, by Eqs. (8.1) and (8.2) and using Table A (Appendix A), we obtain

8.8. Derive the Neyman-Pearson test, Eq. (8.1 7).

From Eq. (8.1 6), the objective function is
J = (1 - 8) - n(a - a,) = P(Dl ]HI) - R[P(D, (H,) - a,,] (8.29)
where 1 is an undetermined Lagrange multiplier which is chosen to satisfy the constraint a = a,. Now, we
wish to choose the critical region R, to maximize J. Using Eqs. (8.1) and (8.2), we have