2. Tiffany and John decide to meet at the train station. If each person independently arrives at a time uniformly distributed between 1 p.m. to 1:30 p.m., calculate the probability that the first to arrive has to wait longer than 10 minutes. (10%)

3. Let $Y_1 < Y_2 < Y_3$ be the order statistics of a random sample of size $n = 3$ from the exponential distribution with p.d.f. $f(x) = e^{-x}$, $0 < x < \infty$. (1) Find the p.d.f. of the sample median \(Y_{2}\). (5%)

(2) Compute the probability that \( Y_{3} \) is less than two. (5%)

(3) Determine \( P(1 < Y_1) \). (5%)

4. A random sample $X_1, X_2, \ldots, X_n$ of size n is taken from a Poisson distribution with a mean of $\lambda, 0 < \lambda < \infty$. Show that $\bar{X}$ is the minimum-variance unbiased estimator for $\lambda$. (15%)

II. 填充題 (50%) All values are rounded to 4 decimal points. 1. Define \(\Delta p\) as the daily change for the value of a portfolio in one day. Suppose that one-day volatility (standard deviation) of \(\Delta p\) is 3 and \(\Delta p\) from one day to the next day are independent. The estimate of 4-day volatility for \(\Delta p\) is (5%) Suppose that daily changes \(\Delta p\) have first-order correlation with 0.1. The 4-day volatility for \(\Delta p\) will be (5%) 1.

2.

2. ABC Company sells kiwi fruits. Suppose the weight of a kiwi fruit is normally distributed with mean \(21\mathrm{g}\) and standard deviation of \(2\mathrm{g}\). There are 16 kiwi fruits packed in a box. The probability that a box of kiwi fruits weights less than \(340\mathrm{g}\) is (5%) Mary buys 5 boxes of kiwi fruits. The probability that there is only one box of kiwi fruits weights less than 340 g is _____.(5%) ABC Company claims that the average weight of a kiwi fruit is at least 21 g. Mary buys 5 boxes of kiwi fruits and find the average weight of 20.5 g for a kiwi fruit. Is the test for this hypotheses significant at $\alpha=0.05$? _____.(5%) 3.

4.

5.

3. A multiple regression analysis was performed on a random sample with the model \( y_{i} = \beta_{0} + \beta_{1}x_{i1} + \beta_{2}x_{i2} + \beta_{3}x_{i3} + \beta_{4}x_{i4} + \epsilon_{i} \quad i = 1, \ldots, 33 \) If the coefficient of determination is .8 (R²), the F-value for testing \( H_{0}: \beta_{1} = \beta_{2} = \beta_{3} = \beta_{4} = 0 \) is (5%) If the correlation between y and x₁ is .2 and a simple linear regression (using x₁ to predict y) was performed, what is the t-value for testing \( H_{0}: \beta_{1} = 0 \)? (5%) 6.

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