II、計算問答題(每大題 10 分，共 20 分。)

1. Mary has $$T = 40$$ observations on variables $$y$$, $$x_1$$, and $$x_2$$. She estimates the following equations by least squares:

$$y = -2.0 + 3.5x_1 - 1.5x_2, \tag{1}$$

(0.7) (0.3)

Residual sum of squares (SSR)=140, $$R^2 = 0.8$$ where the numbers in parentheses are standard errors. What is the true value of SST in equation (1)? What is the value of $$s^2$$ (estimated error variance.)?

2. Consider the simple regression model, assume to satisfy ideal conditions: $$y_t = \beta x_t + \varepsilon_t \tag{2}$$ We now want to consider another estimator $$\tilde{\beta}$$, defined as follow. You pick two observations at random; say, $$t = 3$$ and $$t = 6$$. $$\tilde{\beta} = (y_6 - y_3)/(x_6 - x_3) \tag{3}$$ (a). Consider the estimator \(\tilde{\beta}\). Whether is \(\tilde{\beta}\) unbiased?

(b). Compared with OLS estimator \(\hat{\beta}\), which estimators is efficiency? Please explain.

Ⅲ、填空題(每格 5 分，共 50 分。) 1. Let random variable $X$ follow a geometric distribution with parameter $p$, that is, $$P(X=x) = pq^{n-1}, \quad p+q=1, \quad x=1, 2, 3, \dots$$ Calculate the probability of $P(X > m)$, where $m$ is a positive integer. (1)

2. Suppose $S_n = X_1 + X_2 + \dots + X_n$, where $E(X_i) = \frac{1}{n}$, $Var(X_i) = \frac{1}{n} - \frac{1}{n^2}$ and $Cov(X_i, X_j) = \frac{1}{n(n-1)} - \frac{1}{n^2}$ for $i \neq j$. Then, $Var(S_n) =$ (2).

3. Suppose that the probability density function of $X$ is $f(x) = 2x \exp(-x^2)$, $0 \le x < \infty$. Find the probability density function of $Y = X^2$, $f(y) =$ (3).

4. Suppose that the continuous random variables $(X, Y, Z)$ have a valid joint probability density function $$f(x, y, z) = Kxyz^2, \quad 0 \le x, y \le 1, \quad 0 \le z \le 3.$$ Then, $E(Y) =$ (4) and $E(Y|X = \frac{1}{2}, Z = 1) =$ (5). Of course, you should find the value of $K$ first.

5. If random variables $X_1, X_2, \dots, X_n$ are i.i.d. $N(\mu, \sigma^2)$, and $S^2$ is their sample variance. Then, the distribution of $\frac{(n-1)S^2}{\sigma^2}$ is (6).

6. Suppose you play a game of chance where you win $w$ dollars with probability $p$ but lose $v$ dollars with probability $1-p$. Denote $X_i$ the random variable. Then, how large should $p$ be so that you, a reasonable person, are willing to play the game? (7)

7. Let $X_1, \dots, X_n, X_{n+1}$ be a random sample from a normal population having an unknown mean $\mu$ and variance 1, and $\bar{X}_n$ be the average of the first $n$ variables. Then, the distribution of $X_{n+1} - \bar{X}_n$ is (8). If $\bar{X}_{200} = 5$, then the 95% confidence interval for $X_{201}$ is (9).

8. In a multiple linear regression, a dependent variable is explained by a constant and 4 independent variables using 35 observations. The $R^2$ of this regression is 0.6. To test the significance of this regression, we can obtain the $F$ statistic equal to (10).

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