Page 284 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 8) DECISION THEORY

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

Equation (8.36) indicates that the statistic

provides enough information about the observations to enable us to make a decision. Thus, it is called the
suficient statistic for the maximum likelihood test.

8.14. Consider the same observations Xi, i = 1, . . . , n, of radar s:ignals as in Prob. 8.13, but now, under
H,, Xi have zero mean and variance go2, while under HI? Xi have zero mean and variance a12,
and aI2 > go2. Determine the maximum likelihood test.

In a similar manner as in Prob. 8.13, we obtain

With a12 - > 0, the likelihood ratio is

and the maximum likelihood test is

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

Note that in this case,

is the sufficient statistic for the maximum likelihood test.

8.15. In the binary communication system of Prob. 8.6, suppose that we have n independent obser-
vations Xi = X(ti), i = 1, . . ., n, where 0 < t1 < . < t, I T.
(a) Determine the maximum likelihood test.
(b) Find PI and PI, for n = 5 and n = 10.
=
(a) Setting I(, 0 and I(, = 1 in Eq. (8.36), the maximum likelihood test is
9 n HI