4. Suppose X1, . . ., Xn are iid exp()). (a) Show that Ti = X(1) /> is a pivotal quantity. Find an exact 95% CI for A based on T1. (b) Find a 95% CI for A based on the normal approximation to the distribution of the pivot T2 = X/1. Find the exact coverage probability of this CI. Soln: Use CLT to show that vn(72 - 1) follows (approximately) N(0, 1). The exact distribution: 2nT2 ~ x2(2n). (c) (STAT 850 only) Find the shortest exact 95% CI based on T2. Describe a numerical method for finding the interval. S Soln: For any a1 E (0, 0.05) and a2 = 0.05 - o1, the exact 95% CI based on 2nT2 ~ x2(2n) is givne by 2n X 2nx x32 (2n) ') xi-a, (2n) , Try a sequence of on to search for the shortest interval. (d) (STAT 850 only) Let 1()) be the log-likelihood function, A be the MLE of A. Let W = 2[1(A) -1(X)]. Show that W is an asymptotic pivotal quantity. Find a 95% CI based on W. Soln: By the asymptotic distribution of A, W follows (approximately) x2(1). From P(x2(1)