110年 - 110 國立中央大學_碩士班招生考試_財務金融學系/乙組(一般生):數理統計#105304

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).