5. Let $X_1, X_2, \dots, X_n$ denote a random sample from a gamma-type distribution with $\alpha = 2$ and $\beta = 1/\theta$. Let $H_0: \theta = 1$, $H_1: \theta > 1$.
(a) Argue that there exists a UMP test for $H_0$ against $H_1$. And determine a statistic, say $Y$, upon which the test may be based. Also indicate the best rejection region. (8 points)
(b) Find the pdf of the statistic $Y$ in part (a). If we want a significant level of 0.05, write an equation that can be used to determine the critical region. Let $\Pi(\theta), \theta \ge 1$, be the power function of the test. Express the power function as an integral. (10 points)