[試題] 105-2 鄭明燕 機率導論 第五次小考
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）︰2017/5/25
考試時限（分鐘）：30分鐘
試題 :
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 ].
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