课程名称︰机率导论
课程性质︰数学系必修
课程教师︰郑明燕
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2017/5/25
考试时限（分钟）：30分钟
试题 :
Quiz 5 (2017/5/25)
1. Let X and Y be independent uniform (0,1) random variables.
(a)(20%) Find the probability density function of Z = min{X,Y} and compute
E[Z].
(b)(15%) Find the joint probability density function of U = X＋Y, V = X－Y.
(c)(10%) Find the probability density function of X＋Y.
2. Suppose that random vector (X,Y) has a joint probability density function
(pdf) given by
－x
╭ e , if 0≦x＜∞, 0≦y≦x,
f(x,y) = │
╰ 0 , otherwise.
(a)(15%) Find the conditional pdf of X∣Y=y for any y＞0.
(b)(15%) Find Cov(X,Y).
3. Let X ,..., X be independent and identically distributed random variables
1 n
2 _ 1 n
having mean μ and variance σ . Define X = —— Σ X and
n i=1 i
2 1 n _ 2
S = ——— Σ (X －X ) .
n－1 i=1 i
_ _
(a)(10%) Find E[X] and Var(X).
2
(b)(15%) Find E[S ].