7.設有三個袋子,袋一含4個紅球,5個白球,袋二含3個紅球,2個白球,袋三 含1個紅球,2個白球,茲隨機抽一個袋子,再自其中隨機抽出一球,觀此球 之顏色,若此被抽中之球為白色,求其來自袋一之機率? (5分)

1. (15%) Assume that Z follows a standard normal distribution, i.e. Z~N(0, 1), σ is a constant, and X = . please compute the value of the expectation given by E[Max(X-1,0)] +E[Min(X-1,0)].

2. (15%) Let X1 and X2 be two independent random variables and both of then follow a standard normal distribution. Let Y = .Please find the probability density function of Y.

3. (10%) Assume that X is a random variable with a probability density function f(x) that is positive and continuous when a ＜ x ＜ b; zero elsewhere. The distribution function F(x) of X may be written  F(x) = 0 when x ≤ a, and F(x) = 1 when x ≥b. Assume that there is a number M of the distribution with F(M)=0.3. Let X1, X2 and X3be a random sample from this distribution. LetY1＜Y2 ＜Y3 denote the order statistics of the sample. Please compute the value of Pr(Y2 ＜ M).

4. (10%) Assume that the joint pdf of X and Y is given as follows:Please compute the value Pr(X+Y ＜1).

 A. 求該生10全猜正確的機率(5分)

B. 求該生猜對恰好8題的機率 (5分)

6.自一套撲克牌中,以不放回方式隨機抽出13張牌,求抽出2張方塊,4張黑 桃3張紅心,4張梅花之機率 (5分)

8.設隨機變數X服從Gamma(r=2，λ=1),且X的的probability  density function   (p.d.f.)為為Gamma 函數,試求隨機變數X的眾數 (10分) 

A. (5分)

B. (5分)

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