Example 7.3 .2 lapter 7 Estimation The Variance of the Posterior Beta Distribution. Suppose that the proportion 6 of defective items in a large shipment is unknown, the prior distribution of 6 is the uniform distribution on the interval [0, 1], and items are to be selected at random from the shipment and inspected until the variance of the posterior distribution of 6 has been reduced to the value 0.01 or less. We shall determine the total number of defective and nondefective items that must be obtained before the sampling process is stopped. As stated in Sec. 5.8, the uniform distribution on the interval [0, 1] is the beta distribution with parameters 1 and 1. Therefore, after )7 defective items and 2 non- defective items have been obtained, the posterior distribution of 6 will be the beta distribution wither = y + 1 and = z + 1. It was shown in Theorem 5.8.3 that the vari- ance of the beta distribution with parameters or and is (MB/[(05 + f3)2(a: + 6 + 1)]. Therefore, the variance V of the posterior distribution of 6 will be = (y+1)(z+1) (y+z+2)2(y+z+3)' Sampling is to stop as soon as the number of defectives y and the number of non- defectives 2 that have been obtained are such that V 5 0.01. It can be shown (see Exercise 2) that it will not be necessary to select more than 22 items, but it is neces- sary to select at least seven items. 4