Page 278 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 81 DECISION THEORY 27 1

Notice that with sample size n = 100, both a and fl have decreased from their respective original values
of 0.1587 and 0.0668 when n = 25.

DECISION TESTS

8.6. In a simple binary communication system, during every T seconds, one of two possible signals
s,(t) and s,(t) is transmitted. Our two hypotheses are
H,: s,(t) was transmitted.
HI: s,(t) was transmitted.
We assume that

so@) = 0 and sl(t) = 1 0 < t < T
The communication channel adds noise n(t), which is a zero-mean normal random process with
variance 1. Let x(t) represent the received signal :

We observe the received signal x(t) at some instant during each signaling interval. Suppose that
we received an observation x = 0.6.
(a) Using the maximum likelihood test, determine which signal is transmitted.
(b) Find P, and P,, .

(a) The received signal under each hypothesis can be written as
H,: x=n
HI: x=l+n

Then the pdf of x under each hypothesis is given by

The likelihood ratio is then given by

By Eq. (8.9), the maximum likelihood test is

Taking the natural logarithm of the above expression, we get

Since x = 0.6 > 4, we determine that signal s,(t) was transmitted.
(b) The decision regions are given by
$)
Ro = {x: x < $1 =(-a, R, = {x: x > $1 =(*, m)