課程名稱︰統計學導論
課程性質︰數學系選修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/3/7
考試時限（分鐘）：50
試題 :
1. Suppose that random vector (X,Y) has a joint probability density function
(pdf) given by
{ 24xy , if 0 ≦ x ≦ 1, 0 ≦ y ≦ 1, 0 ≦ x+y ≦ 1,
f(x,y) = {
{ 0 , otherwise
(a) (10%) Are X and Y indepedent random variables?
(b) (10%) Find the conditional pdf of X|Y = y for any 0 < y < 1.
* 2
(c) (10%) Find g (Y) that minimizes E[(X-g(Y)) ] over functions g on R.
(d) (10%) Find Cov(X,Y).
2
2. (20%) If Z ~ N(0,1), find the probability density function of Z .
3. (20%) Find the joint density of X+Y and X/Y, where X and Y are independent
exponential random variables with parameter λ.
Show that X+Y and X/Y are independent.
4. (20%) Find the approximate mean and variance of Y = √X, where X is a
nonnegative random variable with mean 4 and variance 3.