Question: Explain each part in detail. 4. Suppose X1, . .., Xn are iid exp()). (a) Show that Ti = X(1) /A is a pivotal quantity.

Explain each part in detail.

4. Suppose X1, . .., Xn are iid exp()). (a) Show that Ti = X(1) /A is a pivotal quantity. Find an exact 95% CI for A based on Ti. (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(T2 - 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. Soln: For any al E (0, 0.05) and a2 = 0.05 - a1, the exact 95% CI based on 2nT2 ~ x2(2n) is givne by 2nX 2nX Xa, (2n) xi-a (2n) Try a sequence of on to search for the shortest interval

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