115年 - 113 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#146998

1. (15%) Let \( Z = X + Y \). Derive the CDF of \( Z \).

2. (15%) Compute \( \operatorname{E}(\mathrm{X}|\mathrm{Z}) \)

3. (10%) Compute \( \operatorname{E}[\operatorname{E}(\mathrm{X}|\mathrm{Z})] \) based on your answer in 2.

4. (5%) Compute \( \operatorname{E}(\mathrm{X}) \) based on the pdf of \( \mathrm{X} \).

5. (5%) Suppose you are dealing with an unknown joint pdf \( f(x, y) \). Suppose you only have two random samples of size \( n \): one is \( X_1, X_2, \ldots, X_n \) from the distribution of \( X \), and the other is \( W_1, W_2, \ldots, W_n \) from the distribution of \( W \), where \( W = E(X|Z) \). Can you determine which random sample would allow you to derive a more precise estimator of \( E(X) \)? Provide an explanation for your choice.

a. (5%) Find the limiting distribution of \( \sqrt{n} (\hat{p}_U - p) \) and \( \sqrt{n} (\hat{p}_B - p) \), respectively.

b. (5%) Show that \( \hat{p}_U \) and \( \hat{p}_B \) are the consistent estimators of \( p \).

c. (5%) Find a consistent estimator of \( \frac{1}{p} \).

d. (5%) Use \( n = 1 \) to show that no unbiased estimator of \( \frac{1}{p} \) exists.

a. (5%) Find the shortest length \( 100(1 - \alpha)\% \) confidence interval for \( \theta \) in the class \( C(X) := \left[\frac{X_{(n)}}{\alpha}, \frac{X_{(n)}}{b}\right] \).

b. (5%) Prove that \( \mathrm{P}_{\theta}[\theta' \in C(X)] < 1 - \alpha \) for \( \theta' \neq \theta \).

a. (5%) Find a sufficient statistic of \( \theta \).

b. (5%) Find the maximum likelihood estimator (MLE) of \( \theta \).

c. (5%) Find the method of moments estimator of \( \theta \).

4. (5%) Consider the testing problem $H_0: \theta = \theta_0$ versus $H_1: \theta \neq \theta_0$ based on i.i.d. $X_1, X_2, \ldots, X_n$ from $U(0, \theta)$. Show that the uniformly most powerful (UMP) test with rejection region $X_{(n)} > \theta_0$ or $X_{(n)} \le \theta_0 \alpha^{1/n}$ is a likelihood ratio (LR) test with size $\alpha$.