課程名稱︰機率導論
課程性質︰數學系必修
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）︰2017/6/8
考試時限（分鐘）：30分鐘
試題 :
Quiz 6 (2017/6/8)
1. Let X and Y be independent gamma random variables with respective parameters
(α,λ) and (β,λ).
(a)(15%) Find the joint moment generating function of X and Y.
(b)(15%) Use the moment generating function of X to find E[X] and Var(X).
(c)(15%) Use the moment generating function to find the distribution 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 expectation E[X∣Y].
(b)(15%) Find the conditional variance Var(X∣Y).
3.(25%) Suppose X and Y are joint random variables. Show that
2 2
E[(Y－g(X)) ] ≧ E[(Y－E[Y∣X]) ] for any real-valued function g on R.