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︰2005/01/11 13:20～15:30
試題：
Instructions. You have 110 minutes to complete the examination. This is a
closed-book examination.
1. (70％) Let the joint pdf of random variables X and Y be bivariate normal
1
f(x,y)＝ ───────── ‧
2π(1－ρ^2)^(1/2)
╭ 1 2 2 ╮
exp│ ─────[(x－μ_x) －2ρ(x－μ_x)(y－μ_y)＋(y－μ_y) ]│
╰ 2(1－ρ^2) ╯,
(a) (40％) Show that X＋Y and X－Y are independent random variables.
(b) (30％) Find the distribution of aX＋bY, where a and b are nonzero
constants. For solving (b), you cannot use the fact that aX＋bY is
normally distributed.
2. (80％) Let X_1,...,X_n be a random sample from a population with pdf
a a-1
╭ ──‧x if 0＜x＜θ
f_X(x)＝│ θ^a
╰ 0 otherwise.
Let X_(1)＜...＜X_(n) be order statistics.
(a) (40％) Show that X_(1)∕X_(2), X_(2)∕X_(3),..., X_(n-1)∕X_(n), and
X_(n) are mutually independent random variables.
(b) (40％) Find the distribution of each of them.
3. (30％) Let X be a random variable with F_p,q distribution. Show that
(p∕q)X∕[1＋(p∕q)X] has a beta distribution with parameters p/2 and q/2.
Fact 1. When U_1,...,U_a+b are independent exponential with mean 1,
(U_1＋...＋U_a)∕(U_1＋...＋U_a+b) is a beta distribution with
parameter a and b.
Fact 2. The density function of Gamma(α,β) is
1 α-1 ╭ x ╮
──────‧x exp│- ─│
Γ(α)β^α ╰ β╯.
When α＝1, it is an exponential distribution. When α＝p/2 and
β＝2, it is a chi-squared distribution.
4. (50％) The random variable X has a Poisson distribution with parameter λ,
and the conditional distribution of Y given X is
1
P_Y|X(y|x)＝ ── , y＝1,2,...,x＋1,
x＋1
for each x 屬於 {0,1,...}. Find Corr(X,Y).
5. (70％) Let X_1,...,X_n be a sample of iid random variables from N(0,σ^2)
for some positive constant σ^2. Let
n
S_n＝ Σ|X_i|.
i=1
(a) (30％) Find the mean and variance of S_n.
(b) (20％) Show that cos(S_n∕n)＋sin(S_n∕n) converges in probability to
some constant and identify that constant. 2
(c) (20％) Find constants a_n and b_n for which a_n(S_n－b_n) converges in
distribution to a non-degenerate distribution.
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