4. A random variable X has the following probability density function \( f(X) = \frac{e^{-X}}{(1+e^{-X})^2} \), \( -\infty < X < \infty \) 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 \( x_1, x_2, \dots, x_n \) is a random sample of size n from the following double exponential distribution \( f(x | \beta) = \frac{1}{2\beta} e^{-\frac{|x|}{\beta}} \). Assume that \( x_1 = -3, x_2 = 2, x_3 = 4, x_4 = -1 \), and \( x_5 = 1 \). First derive the maximum likelihood estimator of \( \beta \), 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

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. Let \( \{X_n\} \) be a sequence of random variables indexed by the sample size n. Suppose that \( X_n \) takes two values, 1 and n, with probabilities \( 1 - \frac{1}{n} \) and \( \frac{1}{n} \), respectively. Which of the following is correct? (A) \( X_n \) converges in probability. (B) \( X_n \) converges in mean square. (C) \( X_n \) converges in both probability and mean square. (D) \( X_n \) converges in neither probability nor mean square.

1. Consider the linear model $yi = \beta0 + \beta1 x{1i} + \beta2 x{2i} + ui$, where $E(ui \mid x{1i}, x{2i}) = 0$ and $Var(ui \mid x{1i}, x{2i}) = \sigma^2$ (homoskedasticity). Let $R1^2$ be the $R^2$ from the auxiliary regression of $x{1i}$ on $(1, x{2i})$. Suppose $\hat{\sigma}^2 = 4.00$, $\sum{i=1}^{n} (x{1i} - \bar{x}1)^2 = 200$, and $R1^2 = 0.75$. Using the standard homoskedastic OLS variance formula, what is the implied standard error of $\hat{\beta}_1$?(A) 0.071(B) 0.100(C) 0.283(D) 0.400

2. Consider $yi = \beta0 + \beta1 Di + \beta2 xi + \beta3 (Di xi) + ui$, where $Di \in \{0, 1\}$ and $xi \in \mathbb{R}$, with $E(ui \mid Di, xi) = 0$. Define the conditional mean $\mu(D, x) = E(yi \mid Di = D, xi = x)$. Which statement is correct?(A) \( \beta_{1} = \mu (1,x) - \mu (0,x) \) for all \( x \)(B) The marginal effect of \( x \) when \( D = 1 \) is \( \beta_{2} \)(C) \( \beta{3} = \frac{\partial\mu(1,x)}{\partial x}\Big|{x = 0} - \frac{\partial\mu(0,x)}{\partial x}\Big|_{x = 0} \)(D) \( \beta{2} + \beta{3} \) equals the intercept difference between \( D = 1 \) and \( D = 0 \)

3. The true model is $yi = \beta0 + \beta1 xi + \beta2 zi + ui$, $E(ui \mid xi, zi) = 0$, but you estimate the short regression $yi = \alpha0 + \alpha1 xi + ei$ (omitting $zi$). Assume the standard OVB formula:$$plim(\hat{\alpha}1) = \beta1 + \beta2 \frac{Cov(xi, zi)}{Var(xi)}.$$Suppose $\beta1 = 1.0$, $\beta2 = -2.0$, $Var(xi) = 4.0$, and $Cov(xi, zi) = 1.0$. Which is closest to $plim(\hat{\alpha}1)$?(A) 0.0(B) 0.5(C) 1.0(D)-0.5

4. Consider the simple regression $yi = \beta0 + \beta1 xi + u_i$. Which condition, if violated alone, implies OLS is generally not BLUE but can remain unbiased?(A) \( E(u{i}\mid x{i}) = 0 \)(B) \( \operatorname{Var}(ui \mid xi) = \sigma^2 \) (constant variance)(C) \( \operatorname{Var}(x_i) > 0 \)(D) Linear-in-parameters conditional mean: \( E(y{i} \mid x{i}) = \beta{0} + \beta{1}x_{i} \)

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