課程名稱︰高等統計推論
課程性質︰選修∕數研所統計組必修
課程教師︰陳宏 教授
開課系所︰數學系
考試時間︰
試題：
Instructions. You have 60 minutes to complete the examination. This is a
closed-book examination.
1. Suppose there are n balls and n boxes. Suppose that those n balls are
dropped randomly into those n boxes. Let X denote the number of balls
falling into a particular box.
(a) (15%) Determine the distribution of X.
(b) (15%) Find a good approximation of the distribution of X when n is
large.
2. (30%) For x≧0, let [x] denote the integer part of x; e.g. [.5]＝0,
[1]＝1, and [1.3]＝1. Suppose the cdf of a r.v. X is
[x]
F(x)＝1－r , x≧0
where r is some constant in (0,1). Find E(X).
3. Let X be a random variable with the pdf f(x)＝exp(-x)‧I_(0,∞)(x).
2
(a) (15%) Find the pdf of Y＝(X－1) .
(b) (15%) Find a function g such that P(g(X)≦y)＝y, y 屬於 (0,1).
(c) (10%) Find a function g such that
P(g(X)≦y)＝╭ 0, y 屬於 (0,1/2)
╰ y, y 屬於 [1/2,1).
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