108年 - 108 國立臺北大學_碩士班一般入學考試試題_經濟學系：統計學#146890

1. A couple has the same chance to give birth to a boy and a girl, and decides to have 3 children. If none of the 3 is a girl, they will try again; and if they still don't get a girl, they will try once more. If the random variable \( X \) denotes the number of children the couple will have following this scheme, then what is the expected value of \( X \)?      (1)     

2. Let \( X \) be a random variable with density function \( f(x) = 1 - |x|, |x| < 1 \). Then, the variance of \( X \), \( Var(X) \), equals      (2)     .

3. Let the joint probability density function of the continuous random variables \( X \) and \( Y \) be
\( f(x, y) = \frac{6}{5}(x^2 + 2xy) \), \( 0 < x, y < 1 \). Then, \( P(X \leq Y) = \)=       (3)     .

4. Let \( X_{1} \) and \( X_{2} \) be two independent random variables with identical probability density function \( f(x_{i}) = e^{-x_{i}}, x_{i} > 0 \). Then, the probability density function of \( W = \min \{X_{1}, X_{2}\} \), is      (4)     .

5. If \( X_{1}, X_{2}, \ldots, X_{n} \) are normal random variables with variance \( \sigma^{2} \) and covariance between any pair of these random variables is \( \rho \sigma^{2} \), then the variance of \( \frac{1}{n}(X_{1} + X_{2} + \dots + X_{n}) \) equals      (5)     .

6. Suppose \( X \) and \( Y \) are random variables with \( E(Y|X = x) = -X + 3 \) and \( E(X|Y = y) = -\frac{1}{4} Y + 5 \). Then, the correlation coefficient between \( X \) and \( Y \), \( \operatorname{Corr}(X,Y) \), equals      (6)     .

7. Let \( Y = X_{1} + X_{2} + \dots + X_{15} \) be the sum of a random sample of size 15 from the distribution whose density function is \( f(x) = 1.5x^{2}, -1 < x < 1 \). What is the approximate value of \( P(-0.3 \leq Y \leq 1.5) \) when one uses the central limit theorem?      (7)     .

8. Let \( X_{1} \) and \( X_{2} \) be a random sample of size 2 from the standard normal distribution, then the \( 75^{\text{th}} \) percentile of the statistic \( W = \frac{X_{1}}{\sqrt{X_{2}^{2}}} \) equals      (8)     .

(9)

(10)

(1)

(2)

(3)

(4)

(5)

(6)

III. 計算與申論題 (20%。每題 10 分，共 2 題) 請在答案卷標示大題「III」及其小題號 (1) 至 (2)，並陳述對應答案。陳述過程中若有明確數值者，請寫下該數值並四捨五入進位至小數第四位。
(1) (10%) Consider the following regression model with a lagged dependent variable:
$$y_t = \beta_1 + \beta_2 x_t + \beta_3 y_{t-1} + \varepsilon_t,$$
in which the regression residual is suspected to be an AR(1) process. That is, $\varepsilon_t = \rho \varepsilon_{t-1} + v_t$, where $0 < |\rho| \le 1$ and $v_t$ is white noise. The OLS regression output for this model gives you the Durbin-Watson statistic equaling 0.39 from a sample size $T = 50$, Can you test the existence of an AR(1) process in $\varepsilon_t$? If you can, state your test steps and conclusion. If you cannot, state your reason(s)?

(2) (10%) Suppose the true wage equation was $y_i = \beta_1 + \beta_2 x_{2,i} + \beta_3 x_{3,i} + \varepsilon_i$, where individual i's wage (y) is a linear function of education ($x_2$) and ability ($x_3$). $\varepsilon_i$ is white noise. Normally, "ability" is a variable that can hardly be qualified, so Peter estimates a false wage equation as $y_i = \alpha_1 + \alpha_2 x_{2,i} + v_i$, where $v_i$ is assumed to be white noise. Assuming that "ability" is positively correlated with wage and with education, then, is Peter's estimated $\hat{\alpha}_2$ unbiased, positively biased, or negatively biased? Why?