ii. (2 分) 試求條件期望值 E[?|? > 2]。

iii. (3 分) 試求條件變異數 Var[?|? > 2]。

4. (10 分) 假設 ?1, … , ?? ??? ~ ?(?, ? 2 ) 且 ? ∈ ℝ & ? 2 ∈ (0, ∞)。令 ?̅ ? = 1 ?∑?? ? ?=1 ?? 2 = 1 ? − 1∑(?? − ?̅ ? ) 2 ? ?=1 試證 √?(?̅?−?) ?? ? → ?(0,1) 當 ? → ∞。

二、 統計部分： 5. (10 分) Suppose ?1, ?2, ⋯ , ?? are i.i.d. random variables following the exponential distribution with PDF ?(?; ?) = λ? −?? , ? ≥ 0, ? > 0. Find the MLE of λ.

6. Let ?1, ?2, ⋯ , ?? be i.i.d. random variables from a normal distribution N(μ, σ 2 ) where μ is unknown and σ 2 is known. (a) (10 分) Find an unbiased estimator for μ.

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

7. A researcher wants to test if the mean weight of a population of oranges is 150 grams. A random sample of n = 10 oranges is collected, with the sample mean weight ?̅ = 145 grams and sample standard deviation s = 10 grams. (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.

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