Suppose that the proportion p of defective items in a large manufactured lot is unknown, and it is desired to test the following simple hypotheses: H0: p = 0.3, H1: p = 0.4. Suppose that the prior probability that p = 0.3 is 1/4, and the prior probability that p = 0.4 is 3/4; also suppose that the loss from choosing an incorrect decision is 1 unit, and the loss from choosing a correct decision is 0. Suppose that a random sample of n items is selected from the lot. Show that the Bayes test procedure is to reject H0 if and only if the proportion of defective items in the sample is greater than log 7 6 + 1 n log 1 3 log 14 9

Lecture 17 Nicole Rubenstein October 31, 2017 Multivariate distribution De▯nition 1.1. Let Y ;Y ;:::;Y be discrete random variables. The joint probability mass functions 1 2 n for Y1;Y2;:::;Ynare p(y1;y 2:::;y n = P(Y =1y ;Y1= 2 ;::2;Y = y n; n y12 S 1y 2 S ;:2:;y 2 n : n Theorem 1.1. If Y ;1 ;:2:;Y arn discrete random variable with joint probability mass function p(y1;y2;:::;y n, then 1. p(y1;y2;:::;yn) ▯ 0 for all 1 2 S1;y22 S ;2::;y 2nS : n P P P 2. ▯▯▯ p(y ;y ;:::;y ) = 1: y12S1 y22S2 yn2Sn 1 2 n Example 1.1. A local supermarket has three checkout counters. Two c