(c) Assume that the prior distribution on \( p \) is beta \((\alpha, \beta)\).

(c-i) \((3\%)\) Find the posterior distribution of \(p\)

(c-ii) \((3\%)\) Calculate the posterior mean and variance.

(c-iii) \((3\%)\) Denote \(\hat{p}B\) to be the posterior mean found in (c-ii). Find \(\alpha, \beta\) such that \(R(p, \hat{p}B)\) is constant.

(d) Consider three estimators $$\hat{p}_U = \text{UMVUE found in (b)},$$ $\hat{p}_B$ = the posterior mean with constant risk found in (c-iii), and $$\hat{p}T = \begin{cases} \hat{p}U \text{ with probability } \frac{n}{n+1}, \\ 1/2 \text{ with probability } \frac{1}{n+1}. \end{cases}$$ (d-i) \((3\%)\) Find \(\sup{p\in (0,1)}R(p,\hat{p}U)\) \(\sup{p\in (0,1)}R(p,\hat{p}B)\) , and \(\sup{p\in (0,1)}R(p,\hat{p}T)\) for a fixed \(n\)

(d-ii) \((3\%)\) Find the limiting distribution of \(\sqrt{n} (\hat{p}U - p)\) and \(\sqrt{n} (\hat{p}B - p)\), respectively.

(d-iii) \((3\%)\) Show that \(\hat{p}U\) and \(\hat{p}B\) are the consistent estimators of \(p\).

(d-iv) \((3\%)\) Show that \(\hat{p}_U\) is asymptotically efficient of \(p\).

(d-v) \((3\%)\) Find the asymptotic relative efficiency (ARE) of \(\hat{p}U\) with respect to \(\hat{p}B\).

4. (12%) Let X₁, X₂, ..., Xₙ be a random sample from N(μ, σ²), where both parameters are unknown. (a) \((4\%)\) Find a pivotal quantity of \(\mu\)

(b) \((4\%)\) Use the pivotal quantity of (a) to find a \(100(1 - \alpha)\%\) confidence interval (C.I.) for \(\mu\)

(c) \((4\%)\) Is your finding in (b) the shortest length C.I.? Prove your claim.

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