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Exercise 27 (#7.39). Let (X1, ..., Xn) be a random sample from the exponential distribution on (a,0) with scale parameter 1, where a E R is unknown. Find a confidence interval for a having the shortest length within the class of confidence intervals [X (1) + c, X(1) + d) with confidence coefficient 1 - a, where X(1) is the smallest order statistic. which yields c = n-1 log a. The shortest length confidence interval is then [X(1) + n-l log a, X(1)]. 1 Exercise 28 (#7.42). Let (X1,..., Xn) be a random sample from a dis- tribution with Lebesgue density 0x011(0,1)(x), where 8 >0 is unknown. (i) Construct a confidence interval for 0 with confidence coefficient 1 - a, using a sufficient statistic. (ii) Discuss whether the confidence interval obtained in (i) has the shortest length within a class of confidence intervals. (iii) Discuss whether the confidence interval obtained in (i) is UMAU