105年 - 105 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146121

(a) Determine the p.d.f.’s of U and V.(5分)

(b) Show that U and V are independent.(5分)

(c) Compute the probability P(U < 0, V > 0).(5分)

(a) Calculate the E(X) from p.d.f. f (x).(5分)

(b) Determine the moment generating function (m.g.f.) MX of X and specify the range of its argument. (5分)

(c) Employ the m.g.f. in order to derive the E(X2).(5分)

3. 令 (X, Y) 為一隨機向量，其聯合機率密度函數為\[ f(x, y) = \begin{cases} c(x + 2y), & 0 < y < 1, \ 0 < x < 2, \\ 0, & \text{otherwise}, \end{cases} \]

試求 (a) c 的值；　(b) 的邊際分配 (marginal distribution)；　(c) 和 的聯合累積分配函數 (joint cumulative distribution function)。(20分)

1. 一項研究估計某城市選民對核能計劃之支持程度，假設在500名隨機抽樣的選民中只有175名支持此一計劃，試求此城市所有選民支持此核能計劃的95%信賴區間。(15分)

2. A researcher wishes to see if the mean number of days that a basic, low-price, small automobile sits on a dealer’s lot is 29. A sample of 30 automobile dealers has a mean of 30.1 days for basic, low-price, small automobiles. At α = 0.05, test the claim that the mean time is greater than 29 days. The standard deviation of the population is 3.8 days.(10分)

3. A survey found that the average hotel room rate in Taipei is $88.42 and the average room rate in Chiayi is $80.61. Assume that the data were obtained from two samples of 50 hotels each and that the standard deviations of the populations are $5.62 and $4.83, respectively. At α = 0.05, can it be concluded that there is a significant difference in the rates?(10分)

4. 假設我們從兩個獨立常態母體取得樣本變數資料如下：

在顯著水準 \( \alpha = 0.05 \) 下，試檢定 \( H_0 : \mu_x = \mu_y \) 對 \( H_1 : \mu_x \neq \mu_y \)。(15分) Table : Area \( \Phi(x) \) under the standard normal curve to the left of \( x \)