Page 276 - Probability, Random Variables and Random Processes

268 DECISION THEORY [CHAP. 8

for all R, # RT. In other words, RT is the critical region which yields the minimum Bayes' risk for the
least favorable P(Ho). Assuming that the minimization and maximization operations are interchange-
able, then we have
min max C[P(H,), R,] = max min e[P(Ho), R,]
RI PWo) PWo) RI
The minimization of C[P(H,), R,] with respect to R, is simply the Bayes' test, so that
min C[P(H,), R,] = C*[P(H,)]
R 1
where C*[P(H,)] is the minimum Bayes' risk associated with the a priori probability P(H,). Thus, Eq.
(8.25) states that we may find the minimax test by finding the Bayes' test for the least favorable P(Ho),
that is, the P(H,) which maximizes C[P(H,)].

Solved Problems

HYPOTHESIS TESTING
8.1. Suppose a manufacturer of memory chips observes that the probability of chip failure is p = 0.05.
A new procedure is introduced to improve the design of chips. To test this new procedure, 200
chips could be produced using this new procedure and tested. Let r.v. X denote the number of
these 200 chips that fail. We set the test rule that we would accept the new procedure if X s 5.
Let
H,: p = 0.05 (No change hypothesis)
H, : p c 0.05 (Improvement hypothesis)
Find the probability of a Type I error.
If we assume that these tests using the new procedure are independent and have the same probability
of failure on each test, then X is a binomial r.v. with parameters (n, p) = (200, p). We make a Type I error if
X 5 5 when in fact p = 0.05. Thus, using Eq. (2.37), we have

Since n is rather large and p is small, these binomial probabilities can be approximated by Poisson prob-
abilities with IZ = np = 200(0.05) = 10 (see Prob. 2.40). Thus, using Eq. (2.100)' we obtain

Note that H, is a simple hypothesis but H, is a composite hypothesis.

8.2. Consider again the memory chip manufacturing problem of Prob. 8.1. Now let

H,: p = 0.05 (No change hypothesis)
H, : p = 0.02 (Improvement hypothesis)
Again our rule is, we would reject the new procedure if X > 5. Find the probability of a Type I1
error.