112年 - 112 國立臺北大學_碩士班一般入學考試試題_統計學系：數理統計#147018

(a) \( (5\%) \) Compute \( E[Y|X = x] \)

(b) \( (10\%) \) Find \( P[2X\leq 3Y] \) and \( \mathrm{P}[1\leq X + Y\leq 2] \)

(c) \( (5\%) \) Find the pdf of \( Z = X + Y \)

(a) \( (5\%) \) Let \( Y{1} = \min \{X{1},\dots,X{n}\} \). Show that \( Y{n} \) is a consistent estimator of \( \theta \).

(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 limiting distribution of \( Z{n} = n(Y{1} - \theta) \)

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

(e) \( (5\%) \) Find an unbiased estimator of \( \theta \)

(f) \( (5\%) \) Find the MLE of the median of the distribution.

(a) \( (3\%) \) Find a minimal sufficient statistic of \( p \)

(b) \( (3\%) \) Find the uniformly minimum variance unbiased estimator (UMVUE) of \( p \)

(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-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 \( \hat{p}B \).

(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.

(a) (4%) Find a uniformly most powerful (UMP) test of size \( \alpha \in (0, \frac{1}{4}) \) for \( H_0: P = P_0 \) versus \( H_1: P = P_1 \), where \( P_0 \) is \( N(0,1) \) and \( P_1 \) is the double exponential distribution with the p.d.f. \( \frac{1}{4}e^{-|x|/2} \). [Hint: \( \int_{1}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-x^2/2} dx \approx 0.1587 \)]

(b) (4%) Find the power function of the test in (a).