2. (50 points) Suppose Y1, . .., Y,, is a random sample of size n = 6 from an Exp(0) distribution, with mean 0 > 0, i.e., with pdf f ( U10 ) = -exp {-4], y > 0 . (a) Suppose you would like to test Ho : 0 = 4 vs. Ha : 0 = 2 at the significance level a = 0.05. Derive the rejection (critical) region of most powerful test for this testing situation. (b) Under the alternative in (a), derive the power of the test. Further, find S, the probability of the type II error. (c) Show that OML = Y is the ML estimator for 0 (no need to check second derivative). Find a pivotal quantity for 0 based on the ML estimator Y and provide its distribution. (d) Find the 100(1 - a)% two-sided confidence interval for 0 based on the pivotal quantity in part (a)