107年 - 107 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#147279

科目：研究所、轉學考(插大)◆統計學 | 年份：107年 | 選擇題數：0 | 申論題數：11

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所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (0)

申論題 (11)

一、(10%) Let $X1, X2$, and $X3$ be independent and have uniform distribution $U(0, 1)$. The joint p.d.f. is $f(x1, x2, x3) = 1$, where $0 \le x1, x2, x3 \le 1$. What is the probability $P(X1 + X2 + X3 \le 1)$? (Please show the details)
二、(10%) There are two random variables $X$ and $Y$. Given $Y, X$ follows the conditional distribution $f(x|y) = e^{-y} y^x / x!$, where $x = 0, 1, 2, \dots$. Moreover, $Y$ follows the distribution $f(y) = \lambda e^{-y\lambda}$, where $y > 0$ and $\lambda > 0$. What are expectation $E(X)$ and the variance $\text{Var}(X)$? (Please show the details)
三、(10%) The moment-generating function of $X$ is $M(t) = (0.25 + 0.75e^t)^{16}$. What are the p.d.f. of $X$ as well as the expectation $E(X)$ and variance $\text{Var}(X)$? (Please show the details)
四、(10%) Let $X1, X2, \dots, Xn$ be a random sample from $N(\mu, \sigma^2)$ with known mean $\mu$. Moreover, Let $\chi^2{\mathfrak{p}(n)}$ denote the $100p^n$ percentile of $\chi^2(n)$, the Chi-Square distribution with $n$ degrees of freedom, where $0 \le p \le 1$. Namely, $P(Y \le \chi^2_{\mathfrak{p}(n)}) = p$, where $Y \sim \chi^2(n)$. Describe how you would construct a $100(1-\alpha)\%$ confidence interval for the unknown variance $\sigma^2$, where $0 \le \alpha \le 1$.
五、(10%) Let $f(x; \theta) = (1/\theta^2)xe^{-x/\theta}$, where $x > 0$ and $\theta > 0$. Let $X1, X2, \dots, X_n$ denote a random sample of size $n$ for this distribution. What is the maximum likelihood estimator of $\theta$. (Please show the details)
六、X and Y represent the results of throwing two fair dice respectively. Please calculate:

(一)$E(X - Y)$. (3 %)
(二)$Var(X - Y)$. (6 %)
(三)If you have known that $X+Y \le 3$, calculate $Var(X/Y)$. (7 %)
七、X and Y are two random variables, please derive: (12 %)

$$Var(X) = E[Var(X|Y)] + Var(E[X|Y])$$
八、X-Uniform(0,1) and Y-Uniform(0,1), X and Y are independent. Please derive the probability density functions of the following equations:

(一)$W = X^2$ (10 %)
(二)$W = XY$. (12 %)