(b) \((10\%)\) Use the moment generating function technique to find the joint distribution of \(W_{1} = X_{1} + 2X_{2} + X_{3}\)

$W_2 = 2X_1 - X_3$, and $W_3 = -2X_2 + X_3$.

(c) (5%) Compute $P[W_1 > W_2 + W_3]$, where $\sqrt{2} = 1.414$, $\sqrt{3} = 1.732$, $\sqrt{7} = 2.646$, $\sqrt{11} = 3.167$.

B. Let $X_1, X_2, \ldots, X_n$ be a random sample drawn from the pdf $f(x) = \frac{1}{\theta}, 0 < x < \theta; 0 < \theta$ and $F(x)$ denotes the corresponding distribution function. (a) \((5\%)\) Let \(Y_{n} = \max \{X_{1},\dots,X_{n}\}\). Find the limiting distribution of \(Z_{n} = n(1 - Y_{n})\).

(b) \((5\%)\) Let \(Y_{2}\) denote the second smallest item from this sample. Find the limiting distribution of \(W_{n} = nF(Y_{2})\).

(c) (5%) Find the maximum likelihood estimator (MLE) \(\theta_{n}\) of \(\theta\).

(d) (5%) Prove that \(\theta_{n}\) is a consistent estimator of \(\theta\).

(e) (5%) Find an unbiased estimator \(\tilde{\theta}_n\) of \(\theta\).

(f) (5%) Find the MLE of the median of the distribution for this sample.

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 $\alpha$.

(b) (7%) Find the uniformly most accurate (UMA) \(1 - \alpha\) confidence interval. Prove your claim.

(c) (7%) Is your finding in Problem b the shortest length confidence interval (C.I.)? Prove your claim.

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