Page 281 - Probability, Random Variables and Random Processes

CHAP. 81 DECISION THEORY 273

To maximize J by selecting the critical region R,, we select x E R, such that the integrand in Eq. (8.30) is
positive. Thus R, is given by

and the Neyman-Pearson test is given by

and 1 is determined such that the constraint

is satisfied.

8.9. Consider the binary communication system of Prob. 8.6 and suppose that we require that a =
P, = 0.25.
Using the Neyman-Pearson test, determine which signal is transmitted when x = 0.6.
Find PI,.
Using the result of Prob. 8.6 and Eq. (8.1 7), the Neyman-Pearson test is given by

Taking the natural logarithm of the above expression, we get

The critical region R, is thus
R, = {x: x > $ + In A]
Now we must determine A such that a = P, = P(D, ( H,) = 0.25. By Eq. (8.1), we have

+
Thus 1 - @(i In A) = 0.25 or Q(4 + In A) = 0.75
From Table A (Appendix A), we find that Q(0.674) = 0.75. Thus

Then the Neyman-Pearson test is

Since x = 0.6 < 0.674, we determine that signal so(t) was transmitted.
By Eq. (8.2), we have

8.10. Derive Eq. (8.21).