3. Let X and Y be random variables with the following joint probability density function. f(X, Y) = e^{-X}, 0 < Y < X

First derive E(Y | X), and then evaluate it at X = 2. What is the resulting expected value?
(A) 1/2
(B) 1
(C) 2
(D) None of the above

4. A random variable X has the following probability density function. f(X) = e^{-X} / (1 + e^{-X})^2, -∞ < X < ∞Let Y = (1 + e^{-X})^{-1}. Which of the following distributions does the probability density function of Y, f(Y), belong to?(A) Uniform distribution(B) Normal distribution(C) Exponential distribution(D) Poisson distribution

5. Suppose x1, x2, ..., xn is a random sample from a normal distribution with mean μ and variance σ², where both μ and σ² are unknown. Assume x1 = 2, x2 = 4, x3 = 4, x4 = 6, x5 = 8, x6 = 8, and x7 = 10. What is the 90% confidence interval for σ²? Hint: χ²₀.₀₅.₆ = 1.635, χ²₀.₀₅.₇ = 2.167, χ²₀.₉₅.₆ = 12.592, and χ²₀.₉₅.₇ = 14.067.(A) (3.81, 29.36)(B) (4.45, 34.25)(C) (3.41, 22.15)(D) (3.98, 25.84)

6. Consider the hypothesis test for a normal distribution with mean μ at significance level α: H0: μ = μ0 versus H1: μ ≠ μ0. Which of the following is correct?(A) \( \Pr \) (critical region \( |\mathrm{H}_0\rangle = 1 - \alpha \)(B) \( \Pr \) (critical region \( |\mathrm{H}_1\rangle = \Pr \) (type II error)(C) \( \Pr \) (reject \( \mathrm{H}0\mid \mathrm{H}0 \) ) \( = \) p-value(D) \( \Pr \) (reject \( \mathrm{H}0\mid \mathrm{H}1 \) ) \( = \) power

7. Suppose x1, x2, ..., xn is a random sample of size n from the following double exponential distribution. f(x | β) = 1/(2β) e^{-|x|/β}. Assume that x1 = -3, x2 = 2, x3 = 4, x4 = -1, and x5 = 1. First derive the maximum likelihood estimator of β, and then evaluate it at the given sample values. What is the resulting estimate?(A) 0.6 (B) 1.67 (C) 2.2 (D) 0.45試題隨卷繳交接背面2國立臺北大學 115 學年度碩士班一般入學考試試題系(所)組別：經濟學系科 目：統計學第2頁 共4頁☑可 ☐不可使用計算機

8. Suppose $x1, x2, \ldots, xn$ is a random sample of size $n$ from a Poisson distribution with mean $\lambda$, and assume $n = 2k$ for some integer $k$. Consider the following estimator. $\hat{\lambda} = \frac{1}{2k} \sum{i=1}^{k} (x{2i} - x{2i-1})^2$ Which of the following is correct?(A) \( \hat{\lambda} \) is a linear and an unbiased estimator.(B) \( \hat{\lambda} \) is a linear but biased estimator.(C) \( \hat{\lambda} \) is not a linear but an unbiased estimator.(D) \( \hat{\lambda} \) is neither a linear nor an unbiased estimator.

9. Suppose $x1, x2, \ldots, x_n$ is a random sample of size $n$ from a normal distribution with mean $\mu$ and variance $\sigma^2$. What is the Cramér-Rao lower bound for $\sigma^2$?(A) $\frac{\sigma^2}{n}$(B) $\frac{\sigma^2}{n-1}$(C) $\frac{2\sigma^4}{n}$(D) $\frac{2\sigma^4}{n-1}$

10. Consider the wage equation $\ln(wagei) = \beta0 + \beta1 educi + ui$, where $ui$ contains unobserved ability $abilityi$ and local labor-market quality $Li$. Assume $Cov(educi, abilityi) > 0$ and $Li$ directly affects wages. A proposed instrument is distance to nearest college, $zi$. Suppose $zi$ is strongly correlated with $Li$ (e.g. urban areas have smaller distance and higher wages even holding education fixed), and $L_i$ is omitted from the regression. Which statement is most correct?(A) The exclusion restriction may fail because \( z{i} \) can affect \( \ln (wage{i}) \) through \( L{i} \), so \( Cov(z{i},u_{i}) \) may be nonzero.(B) If \( Cov(z{i}, e d u c{i}) \neq 0 \), IV is always consistent regardless of \( L_{i} \).(C) Adding an intercept is sufficient to control for \( L_{i} \)(D) If OLS is biased, IV must be unbiased in finite samples

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