2. (10%) Let \( X \) have an exponential distribution with mean \( \theta > 0 \). Show that \( P(X > x + y | X > x) = P(X > y) \).

3. (15%) Let \( X{1}, X{2} \) be a random sample of size \( n = 2 \) from a distribution with p.d.f. \( f(x) = 4x^{3} \), \( 0 < x < 1 \), zero elsewhere. Given the ratio \( Y = X{1} / X{2} \), find (1)(10%) E(Y). (Round the answer to 4 decimal places)

(2)(5%) \(\operatorname{Var}(Y)\). (Round the answer to 4 decimal places)

4. (10%) A die was cast $n = 120$ independent times and the following data resulted: | Spots up | 1 | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | --- | | Frequency | b | 20 | 20 | 20 | 20 | 40-b | If we use a chi-square test, for what values of b would the hypothesis that the die is unbiased be rejected at the 0.025 significance level? $$(\chi^2{(5),\alpha=0.025} = 12.8; \chi^2{(6),\alpha=0.025} = 14.4; \chi^2{(5),\alpha=0.05} = 11.1; \chi^2{(6),\alpha=0.05} = 12.6)$$

5. (15%) An OLS regression has the following data: $\bar{X} = 3$; $\bar{Y} = 4$; $$\Sigma(X - \bar{X})^2 = 32; \Sigma(Y - \bar{Y})^2 = 29; \Sigma(X - \bar{X})(Y - \bar{Y}) = 28;$$ The sample observation size is 10. Please calculate: (1)(5%)The regression line \(\hat{Y} = \hat{\alpha} +\hat{\beta} X\)

(2)(5%)The variance of residuals \(\hat{\sigma}^2\)

(3)(5%)The t-statistic of the test \(H_0: \beta = 0\)

6. (20%) Throwing two fair dice, each has six faces (1,2,3,4,5,6 points). $X1, X2$ represent the number of results respectively. $Y=\max(X1, X2)$. Please calculate: (1) \((10\%)\mathrm{E}(\mathrm{Y})\)

(2) \((10\%)\mathrm{Var}(\mathrm{Y})\)

7. (15%) There are N students in a class. How large N will let the probability of at least two students having the same birthday exceed 90%? Note: A year has 365 days. 【Hint: You may use the following characters: (1) \(e^{-x}\geq 1 - x\) (2) \(e^{-1} = 0.3678;\ln (0.1) = -2.30259\)

(3) \(m \times (m + 1) \times (m + 2) \ldots (m + k) \leq \left(m + \frac{k}{2}\right)^{k + 1}\)

115年 - 110 國立臺北大學_碩士班一般入學考試試題_企業管理學系甲組：統計學#146951

115年 - 115 國立臺北大學_碩士班一般入學考試試題_金融與合作經營學系：統計學#146920

115年 - [無官方正解]115 國立臺北大學_碩士班一般入學考試試題_經濟學系：統計學#146534

115年 - [無官方正解]115 國立嘉義大學_碩士班招生考試試題_財務金融學系：統計學#143862

115年 - 115 中央警察大學_碩士班入學考試試題：統計學#139759

114年 - 114 國立臺北大學_碩士班一般入學考試試題_企業管理學系甲組：統計學#146907

114年 - 114 中央警察大學_碩士班入學考試試題_警察政策研究所、刑事警察研究所、資訊管理研究所、鑑識科學研究所、行政管理研究所、防災研究所、水上警察研究所海洋科技組：統計學(同等學力加考)#146791

114年 - [無官方正解]114 國立嘉義大學_碩士班招生考試試題_財務金融學系：統計學#144015

114年 - [無官方正解]114 臺灣綜合大學系統_學士班轉學生考試試題:統計學#138278

113年 - 113 國立臺北大學_碩士班一般入學考試試題_自然資源與環境管理研究所：統計學#146880