a. Prove the following, which is related to Theorem 9.3.2. Let X ~ f(x), where f is a strictly decreasing pdf on [0, ∞). For a fixed value of 1 - α, of all intervals [a, b] that satisfy ∫ba f(x) dx = 1 - α, the shortest is obtained by choosing α = 0 and b so that ∫ba f (x) dx = 1 - α.
b. Use the result of part (a) to find the shortest 1 - α confidence interval in Example 9.2.13.