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Suppose that the proportion θ of defective items in a large manufactured lot is knownto be either 0.1 or 0.2, and the prior pmf of θ is as follows:ξ(0.1) = a and ξ(0.2) = b. (1)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 θ has mean 0.15 andstandard deviation 0.05.
(b) Determine the posterior pmf of θ using the selected a and b in the previousproblem.
(c) What is the mean and standard deviation of θ using its posterior distributionξ(θ|X = x).
(d) The loss due to defective items is L = 1000 · θ. What is the expected value of L?
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