2. (10 分) 假設連續型隨機變數 ? 且服從以下機率密度函數：
\[
f(x) = \begin{cases}
\frac{1}{2\pi}, & 0 \le x \le 2\pi \\
0, & \text{其他}
\end{cases}
\]
試求 \( Y = X^2 + 2X + 1 \) 之機率密度函數。

i. (5 分) 試求X|X > 2 的條件機率密度函數。

iii. (3 分) 試求條件變異數 Var[X|X > 2]。

4. (10 分) 假設 \( X_1, ..., X_n \stackrel{iid}{\sim} N(\mu, \sigma^2) \) 且 \( \mu \in \mathbb{R} \& \sigma^2 \in (0, \infty) \)。令\[ \overline{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \]\[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \overline{X}_n)^2 \]試證 \( \frac{\sqrt{n}(\overline{X}_n - \mu)}{S_n} \xrightarrow{D} N(0,1) \) 當 \( n \to \infty \)。

二、 統計部分：5. (10 分) Suppose \(X_1, X_2, \dots, X_n\) are i.i.d. random variables following the exponential distribution with PDF\[ f(x; \lambda) = \lambda e^{-\lambda x}, x \ge 0, \lambda > 0. \]Find the MLE of \(\lambda\).

(a) (10 分) Find an unbiased estimator for μ.

(b) (10 分) Show that your estimator is the UMVUE for μ.

(a) (10 分) State the null and alternative hypotheses for a one-sample t-test.

(b) (10 分) Conduct the t-test at a significance level of α=0.05. Include the calculation of the test statistic and the decision rule.

ii. (2 分) 試求條件期望值 E[X|X > 2]。

一、 Critical Thinking and Writing (50%) In recent years, Taiwanese government has launched the Bilingual 2030 policy to boost the competiveness of the next generation in the hope of assisting young people in getting greater employment opportunities both locally and internationally. The use of EMI (English as a medium of instruction) is often thought as a solution to enhancing students' English proficiency and is associated with discourses on internationalization in higher education reform. With more universities regarding EMI as indispensable, the swift expansion of EMI in Taiwan poses many challenges to the implementation of classroom teaching and leads to heated debates and much controversy.  Please write an essay to illustrate your own experience, observation, or argument regarding the advantages or disadvantages of an EMI classroom. You are expected to write at least two paragraphs.

114年 - 114 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144147

110年 - 110 國立臺灣大學_碩士班招生考試_應用數學科學研究所數理統計組:機率統計#102196