課程名稱︰ 高等統計推論(上)
課程性質︰ 選修
課程教師︰ 江金倉
開課學院： 理學院
開課系所︰ 數學系
考試日期（年月日）︰ 98/1/16
考試時限（分鐘）： 180分鐘
是否需發放獎勵金： 是
（如未明確表示，則不予發放）
試題 :
1. State or define the following terms:
(1a) exponential family. (1b)first-order delta method.
(1c) Jensen's inequality. (1d) Holder's inequality.
(1e) ancillary statistic. (1f) sufficient statistic.
(1g) likelihood function. (1h) elliptical distribution.
(1i) strong law of large numbers. (1j) Boole's inequality.
2.State and show the central limit theorem.
3.Let X~Binomial(n,p) and Y~Negative Binomial(r,p). Show that
F_X(r-1) = 1 - F_Y(n-r).
4.Let X and Y be iid N(0,1) random variables. Find the distribution of
2XY/√(X^2+Y^2).
5.Let U_1,…,U_n,… be iid Uniform(0,1) random variables and X have
distribution P(X=x) = (1/[(e-1)x!])I_{1,2,3,…}(x). Find the distribution of
Z = min{U_1,…,U_X}.
n
6.Let U_1,…,U_n,… be iid Uniform(0,1) random variables, and S_n = ΣU_i
i=1
Define N = min{ k: S_k＞1 }. Show that E[N] = e.
7.Let X~f(x) and generate a random sample Y_1,…,Y_n from g(y), which has the
same support of f(x). Moreover, let X* be a random variable with probability
mass function P(X*=Y_k) = q_k with
n
q_k = [f(Y_k)/g(Y_k)]/[Σf(Y_i)/g(Y_i)], k=1,…,n. Show that
i=1
p
P(X*≦x) → P(X≦x) as n→∞.
8.Let X_1,…,X_n be a random sample from the pdf
f(x|u) = exp(-(x-u))I_{(μ,∞)}(x). Show that X_(1) = min{X_1,…,X_n} is
a complete sufficient statistic for μ.
9.Let X_1,…,X_n be a random sample from N(θ,θ^2) with θ＞0. Moreover, let
_
X_n and (S_n)^2 denote the sample mean and sample variance. Show that
_
(X_n,(S_n)^2) is a sufficient for θ, but the family of distribution is not
complete.
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