It is desired to test the null hypothesis that a sequential sample has a U(0, 2)

common density versus the alternative hypothesis that the common density is U(0, 1). The cost of each observation is

c, and the decision loss is "0-K," loss.

Let w, denote the prior probability of H;.

(a) Show that the Bayes sequential procedure is one of the procedures

d, (Ja nonnegative integer) defined as follows. The procedure do is simply the immediate Bayes decision. For J1,

d, starts sampling; stops sampling when stage n = J is reached, deciding Ho if x, ≥ 1 and deciding H, otherwise;

and at stage n

if x, ≥1, stop sampling and decide Ho;

if x <1, continue sampling.

(b) For

d, (with J1), show that = 2, x₁ =0, E[N|H]=2(1-2), and E[N H]= J.

(c) Find the Bayes sequential procedure if K, = 2, K₁ =1, c=13, and o=.

(d) Find the Bayes sequential procedure if K, =2, К₁ = 1, c=13, and пo = 8.