B. Let X_1, X_2, , X_n be a random sample drawn from the pdf f(x) = 1/θ, 0 < x < θ; 0 < θ and $F(x)$ denotes the corresponding distribution function. (a) (5%) Let Y_n = \X_1,,X_n\. Find the limiting distribution of Z_n = n(1 - Y_n).

Table 1: Area under the standard normal curve II. A. Let $X_1, X_2, \ldots, X_n$ be a random sample from a $U(0, \theta)$ population. (a) (7%) Consider $H_0: \theta = \theta_0$ versus $H_1: \theta \neq \theta_0$ and define $X_{(n)} = \max_{1 \leq i \leq n} X_i$. Show that $$T^*(X): \text{Reject } H_0 \text{ if } X_{(n)} > \theta_0 \text{ or } X_{(n)} \leq \theta_0 \alpha^{1/n}$$ is a uniformly most powerful (UMP) test of size $\alpha$.

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