108年 - 108 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#146113

1. 設二項隨機變數 \( Y = \sum_{i=1}^{6} X_i \sim B(20, p) \)，其中 \( \{X_i, i=1,2,\dots,6\} \) 彼此間互相關，且已知 \( X_i \sim B(i, p), i=1,2,3,4,6 \)。試求：\( X_1 + X_3 + X_5 \) 之分配(需註明名稱及參數)，並計算期望值 \( E(X_1 + X_3 + X_5)^2 \)。(10 分)

2. 假設大腸癌的發病年齡 X 服從平均數為 45 歲、標準差為 10 歲的常態分配，
則某人於 35 歲前發病的機率為何？(利用標準常態分配的累積分配函數
(cumulative distribution function) Φ() x 來表示) (10 分)

3. If the random variable $X$ is distributed as $N(\mu, \delta^2)$ and $0 < p < 1$. Show that the $p_{th}$ quantile of X is given by: $x_p = \mu + \delta \cdot \Phi^{-1}(p)$, where the $\Phi(x)$ is the cumulative distribution function of a standard normal random variable. (10 分)

4. Let \( X \) have the probability density function given by: \[ f_X(x) = \begin{cases} \frac{x+1}{2}, & \text{if } -1 \le x \le 1 \\ 0, & \text{elsewhere} \end{cases} \] Find the density function for \( U = X^2 \) and the mean value \( E[U] = ? \) (10 分)

(a) Find the marginal p.d.f.'s \( f_x \) and \( f_y \), and specify the range of the arguments involved.

(b) Calculate the conditional probability \( P(X > 2\ln(2) | Y = \ln(2)) = ? \)

(a) Find the frequency distribution of sample means.

(b) Find the mean of sample means, and the standard deviation of sample
means.

2. A researcher wishes to estimate, with 95% confidence, the proportion of people
who did not have a home computer. A previous study shows that 40% of those
interviewed did not have a home computer. The researcher wishes to be accurate
within 2% of the true proportion. Find the minimum sample size necessary.
(10 分)

3. 若 \( X_1, X_2, \dots, X_n \) 是抽自常態分配 \( N(\mu, \sigma^2) \) 的一組隨機樣本，其中 \( \sigma \) 是已知的，則試求 \( \mu \) 的 95% 最短區間長度之信賴區間為何？ (10 分)

5. 假設某大學應用數學系甲班全體同學身高呈常態分配，陳老師宣稱甲班全體 同學身高變異數為 20(公分) ² ，今隨機抽取甲班 30 位同學，測得其平均身高 為 165 公分，標準差為 5 公分，則試檢定陳老師的宣稱是否值得採信？ (顯 著水準α = 0.02 ) (10 分)