111 年 國立臺灣大學財務金融研究所丙組《商用統計學》

Multiple Choice Questions. Notes:
(1) Please choose only one of the answer choices (a)-(e).
(2) Write down your answers on the scantron answer sheet.
(3) Each question is worth 5 points.

1. Suppose that X is a random variable that has the characteristic function $\phi_X(t) = \exp(-\frac{1}{2}\lambda^2 t^2)$, where $\lambda$ is a real number. Denote $Z_1 := (1+X)^4$ and $Z_2 := \sin(X) + 2$. Which of the following is right?
   a. $E(Z_1) = 8$ and $E(Z_2) = -2$
   b. $E(Z_1) = 10$ and $E(Z_2) = 0$
   c. $E(Z_1) = 12$ and $E(Z_2) = 2$
   d. $E(Z_1) = 14$ and $E(Z_2) = 4$
   e. None of the above choices (a)-(d).

2. Let X be a random variable that has the probability density function:
   $f(x, \zeta) = \frac{\exp(-x/\zeta)}{\zeta(1+\exp(-x/\zeta))^2}$
   where $x \in \mathbb{R}$ and $\zeta > 0$ is the parameter. In addition, we define $Z_1 := \frac{1}{\zeta} \ln f(X, \zeta)$ and $Z_2 := Z_1^2 + \frac{1}{\zeta} \ln f(X, \zeta)$. Which of the following is right?
   a. $E(Z_1) = \zeta\pi$ and $E(Z_2) = \frac{\pi^2}{6}$
   b. $E(Z_1) = 0$ and $E(Z_2) = 0$
   c. $E(Z_1) = \zeta\pi$ and $E(Z_2) = \frac{\pi^2}{6}$
   d. $E(Z_1) = \frac{\pi}{3}$ and $E(Z_2) = \frac{\pi^2}{6}$
   e. None of the above choices (a)-(d).

3. Let $\{Z_i\}_{i=1}^\infty$ be an IID sequence of $N(0, 1)$-distributed random variables. Consider the following two processes:
   with $X_0 = Z_0$, and
   $X_i = X_{i-1} + Z_i, \quad i=1, 2, \dots$
   $Y_i = Z_{i-1} + Z_i, \quad i=1, 2, \dots$
   Define $\bar{X}_n := \frac{1}{n} \sum_{i=1}^n X_i$ and $\bar{Y}_n := \frac{1}{n} \sum_{i=1}^n Y_i$. Please calculate $\text{var}(X_n)$ and $\text{var}(\bar{Y}_n)$.
   a. $\text{var}(X_n) = n$ and $\text{var}(\bar{Y}_n) = \frac{48}{n^2}$
   b. $\text{var}(X_n) = n + 1$ and $\text{var}(\bar{Y}_n) = \frac{4n-6}{n^2}$
   c. $\text{var}(X_n) = n + 1$ and $\text{var}(\bar{Y}_n) = \frac{4n-2}{n^2}$
   d. $\text{var}(X_n) = n$ and $\text{var}(\bar{Y}_n) = \frac{4n-2}{n^2}$
   e. None of the above choices (a)-(d).

4. Let $\{X_i\}_{i=1}^\infty$ be an IID sequence of $N(0, 1)$ random variables, and $\{Z_i\}_{i=1}^\infty$ be another IID sequence of $N(0, 1)$ random variables. In addition, $\{X_i\}_{i=1}^\infty$ is independent of $\{Z_i\}_{i=1}^\infty$, and we define $Y_i := 2X_i + 3Z_i$. Consider a simple linear regression:
   $Y_i = \beta_0 + \beta_1 X_i + e_i,$
   where $\beta_0$ and $\beta_1$ are regression coefficients, and $e_i$ is a zero-mean error. Let $\hat{\beta}_0$ and $\hat{\beta}_1$ be the ordinary least squares estimators for $\beta_0$ and $\beta_1$, respectively. Suppose that $\hat{\beta}_0$ and $\hat{\beta}_1$ are, respectively, consistent for $\beta_0$ and $\beta_1$, as $n \to \infty$. Please calculate $\hat{\beta}_0$ and $\hat{\beta}_1$.
   a. $\hat{\beta}_0 = 9$ and $\hat{\beta}_1 = 6$
   b. $\hat{\beta}_0 = 7$ and $\hat{\beta}_1 = 4$
   c. $\hat{\beta}_0 = 5$ and $\hat{\beta}_1 = 3$
   d. $\hat{\beta}_0 = 1$ and $\hat{\beta}_1 = 1$
   e. None of the above choices (a)-(d).

5. Let $\{Y_i\}_{i=1}^\infty$ be an IID sequence of Student's t(2)-distributed random variables. Denote $\mu := E(Y)$, $\bar{Y}_n := \frac{1}{n} \sum_{i=1}^n Y_i$, $\sigma^2 := \frac{1}{n-1} \sum_{i=1}^n (Y_i - \bar{Y}_n)^2$, and
   $Z_n := \frac{\sqrt{n}(\bar{Y}_n - \mu)}{\sqrt{\sigma^2}}$
   Note that " $\xrightarrow{d}$ " means that "convergence in distribution." Which of the following is right?
   a. $Z_n \xrightarrow{d} N(0,4)$ and $\sigma^2 \xrightarrow{d} \chi^2(4)$, as $n \to \infty$
   b. $Z_n \xrightarrow{d} N(0,3)$ and $\sigma^2 \xrightarrow{d} \chi^2(3)$, as $n \to \infty$
   c. $Z_n \xrightarrow{d} N(0,2)$ and $\sigma^2 \xrightarrow{d} \chi^2(2)$, as $n \to \infty$
   d. $Z_n \xrightarrow{d} N(0, 1)$ and $\sigma^2 \xrightarrow{d} \chi^2(1)$, as $n \to \infty$
   e. None of the above choices (a)-(d).

6. Let $\{X_i\}_{i=1}^\infty$ be an IID sequence of random variables with zero mean and variance $\sigma^2 < \infty$. Define $\bar{X}_n := \frac{1}{n} \sum_{i=1}^n X_i$. Let $C_n(t)$ be the characteristic function of the statistic $n^{1/2}(\bar{X}_n - \mu)/\sigma$, and $C(t)$ be the characteristic function of the random variable $X_i$. We also let $\lambda$ be an arbitrary real number. Which of the following is right?
   a. $C_n(t) = C(\frac{t}{\sqrt{n}})$
   b. $C_n(t) = C(\frac{t}{\sigma\sqrt{n}})$
   c. $C_n(t) = C(\frac{t}{\sigma})\lambda^{-2}$
   d. $n C_n(t) = C(\frac{t}{\sigma})^{-1}$
   e. None of the above choices (a)-(d).

7. Let Y be a random variable with two possible outcomes: 0 and 1, and $X_i$ be a $N(0, 1)$-distributed random variable. Suppose that $\{(Y_i, X_i)\}_{i=1}^n$ is an IID sequence, and $P(Y_i = 1 | X_i) = p(X_i) > 0$. Consider a simple linear regression:
   $Y_i = \beta X_i + e_i,$
   where $e_i$ is a zero-mean error. Let $\hat{\beta}$ be the ordinary least squares estimator for $\beta$, and define $\hat{e}_i := Y_i - \hat{\beta} X_i$ and $\hat{\sigma}^2 := \frac{1}{n-1} \sum_{i=1}^n \hat{e}_i^2$. Suppose that $\hat{\beta}$ and $\hat{\sigma}^2$ are, respectively, consistent for $\beta^*$ and $\sigma^{*2}$, as $n \to \infty$. Please calculate $\beta^*$ and $\sigma^{*2}$.
   a. $\beta^* = E(p^2(X_i))$ and $\sigma^{*2} = E(p(X_i)(1-p(X_i)))$
   b. $\beta^* = E(X_i p(X_i))$ and $\sigma^{*2} = E(p(X_i)(1-p(X_i)))$
   c. $\beta^* = E(X_i p(X_i))$ and $\sigma^{*2} = \beta^{*2} + E(p(X_i))$
   d. $\beta^* = E(p^2(X_i))$ and $\sigma^{*2} = \beta^{*2} + E(p(X_i)(1-p(X_i)))$
   e. None of the above choices (a)-(d).

8. Let $\{(Y_i, X_{1i}, X_{2i})\}_{i=1}^n$ be an IID sequence of trivariate normal random variables. The mean vector and the covariance matrix of $(Y_i, X_{1i}, X_{2i})'$ are unknown. Consider the following linear regression:
   $Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + e_i,$
   where $e_i$ is a zero-mean error. Let $P$ be the p-value of the t statistic for the following hypotheses:
   $H_0: \beta_2 = 0;$
   $H_1: \beta_2 > 0.$
   Please calculate the second moment and the variance of $P$ under $H_0$.
   a. $E(P^2) = 1/5$ and $\text{var}(P) = 1/16$
   b. $E(P^2) = 1/4$ and $\text{var}(P) = 1/14$
   c. $E(P^2) = 1/3$ and $\text{var}(P) = 1/12$
   d. $E(P^2) = 1/2$ and $\text{var}(P) = 1/10$
   e. None of the above choices (a)-(d).

9. Let $\{X_i\}_{i=1}^n$ be an IID sequence of random variables with $\mu = E(X_i)$ and $\sigma^2 = \text{var}(X_i)$, and denote $\bar{X}_n := \frac{1}{n} \sum_{i=1}^n X_i$. Suppose that we have the following inequality:
   $P(|\bar{X}_n - \mu| \ge \delta) \le \frac{\text{var}(\bar{X}_n)}{\delta^2}$
   for an arbitrary constant $\delta > 0$. Which of the following is right as $n \to \infty$?
   a. $\text{var}(\bar{X}_n) = 1$, and we may use this inequality to show the normality of $\bar{X}_n$
   b. $\text{var}(\bar{X}_n) = \frac{\sigma^2}{n}$, and we may use this inequality to show the efficiency of $\bar{X}_n$
   c. $\text{var}(\bar{X}_n) = \frac{\sigma^2}{n}$, and we may use this inequality to show the normality of $\bar{X}_n$
   d. $\text{var}(\bar{X}_n) = \frac{\sigma^2}{n}$, and we may use this inequality to show the consistency of $\bar{X}_n$
   e. None of the above choices (a)-(d).

10. Let $\{Z_i\}_{i=1}^n$ be an IID sequence of logistic random variables with $\mu = E(Z_i)$ and $\sigma^2 = \text{var}(Z_i)$. Denote $\bar{Z}_n := \frac{1}{n} \sum_{i=1}^n Z_i$, $\tilde{Z}_i := Z_i - \bar{Z}_n$, and $\nu_i := Z_i - \mu$. Which of the following is right?
    a. $n^{-1/2} \sum_{i=1}^n \tilde{Z}_i \xrightarrow{d} 0$, as $n \to \infty$
    b. $n^{-1/2} \sum_{i=1}^n \nu_i \xrightarrow{d} 0$, as $n \to \infty$
    c. $n^{1/2}(\bar{Z}_n - \mu) \xrightarrow{d} 0$, as $n \to \infty$
    d. $n(\bar{Z}_n - \mu)^2 \xrightarrow{d} 0$, as $n \to \infty$
    e. None of the above choices (a)-(d).