Suppose that the proportion of defective items in a large manufactured lot is known to be either 0.1 or 0.2, and the prior pmf of is as follows: (0.1) = a and (0.2) = b.

Suppose also that when n items are selected at random from the lot, it is found that exactly X = x of them are defective. We observe that n = 10 and x = 2.

(a) Determine the values of a and b such that the prior pmf of has mean 0.15 and standard deviation 0.05.

(b) Determine the posterior pmf of using the selected a and b in the previous problem.

(c) What is the mean and standard deviation of using its posterior distribution (|X = x)?

(d) The loss due to defective items is L = 1000 . What is the expected value of L?