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 70 minutes to complete the examination. This is a
closed-book examination.
2
1. (25%) It is known that X_i～N(μ,σ ), i＝1,...,n, Z_j～N(0,1),
_ _
j＝1,...,m, and those variables are independent. Let X and Z denote the
sample averages of {X_i} and {Z_i}, respectively. Set S_Z to be the
sample standard deviation of {Z_i}.
Give the distributions of the following random variables:
X_1－X_2
(a) X_2＋2X_3, (b) ───── ,
√2‧σS_Z
n _ 2
(m－1) Σ(X_i－X)
Z_1 i=1
(c) ─────, (d) ──────────.
2 (1/2) 2 m _ 2
(Z_2) σ Σ(Z_j－Z)
j=1
Also give a brief explanation to your answers.
(1/n)
2. (20%) It is known X is uniformaly distributed over [0,1]. Set Z_n＝X ,
n≧1. Find the limit of Z_n as n→∞.
(1/n)
Hint: Think of the limit of c first and guess the possible limit of
Z_n. Then you can derive the distribution of Z_n and argue from there.
3. (35%) We have a sequence of random variables {X_n, n≧1} in which X_n has
a Poisson(nλ) distribution. Here λ＞0.
(a) (15%) Describe the limit of X_n∕n as n→∞ and give reason to support
your answer. ___
(b) (20%) Show that (X_n－nλ)∕√nλ converges in distribution to a
standard normal variable.
4. (30%) If (X,Y) has a bivariate normal pdf
1 ╭ -1 2 2 ╮
f(x,y)＝ ──────── exp│────── (x －2ρxy＋y )│,
2 (1/2) │ 2 │
2π(1－ρ ) ╰ 2π(1－ρ ) ╯
_______
(a) (15%) Determine the distribution of Z_1＝(X－ρY)∕√1－ρ^2 .
(b) (15%) Determine the distribution of
X^2＋Y^2 －2ρXY
V＝ ───────── .
1－ρ^2
Some solutions or hints:
2
1. (a) N(3μ,5σ ).
(b) t_(m-1).
(c) t_1.
(d) (n-1)‧F_(n-1,m-1).
(1/n) n n
2. For 0≦z≦1, P(Z_n≦z)＝P(X ≦z)＝P(X≦z )＝z → 1.
3. (a) X_n∕n → E(U_i)＝λ as n→∞ by LLN.
___
(b) (X_n－nλ)∕√nλ → N(0,1) as n→∞ by CLT.
4. (a) N(0,1).
2
(b) χ_2.
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