Page 286 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 81 DECISION THEORY

Supplementary Problems

8.17. Let (XI, . . . , X,) be a random sample of a Bernoulli r.v. X with pmf

where it is known that 0 < p 5 3. Let

and n = 20. As a decision test, we use the rule to reject H, if I:=, xi s 6.
(a) Find the power function g(p) of the test.
(b) Find the probability of a Type I error a.
(c) Find the probability of a Type I1 error @ (i) when p, = $ and (ii) when p, = &.

Ans. (a) g@)= (2i)pk(l -P)~'-* O<pli
k=O
(b) a = 0.0577; (c) (i) /3 = 0.2142, (ii) /3 = 0.0024

8.18. Let (X,, . . . , X,) be a random sample of a normal r.v. X with mean p and variance 36. Let
H,: p=50
HI: p = 55
As a decision test, we use the rule to accept H, if 2 < 53, where 2 is the value of the sample mean.
(a) Find the expression for the critical region R,.
(b) Find a and @ for n = 16.

(b) a = 0.0228, 8 = 0.0913

8.19. Let (X,, . . . , X,) be a random sample of a normal r.v. X with mean p and variance 100. Let

As a decision test, we use the rule that we reject H, if 22 c. Find the value of c and sample size n such that
a = 0.025 and B = 0.05.
Ans. c = 52.718, n = 52

8.20. Let X be a normal r.v. with zero mean and variance a2. Let
H,: a2 = 1
HI: a2 =4
Determine the maximum likelihood test.
HI
Ans. 1x1 2 1.36
Ho

8.21. Consider the binary decision problem of Prob. 8.20. Let P(H,) ==,$ and P(H,) = 3. Determine the MAP
test.