作者fatsheepgary (老羊)
看板NTU-Exam
标题[试题] 96下 江金仓 统计学 期末考
时间Sun Jul 6 18:07:15 2008
课程名称︰统计学
课程性质︰选修
课程教师︰江金仓
开课学院:理学院
开课系所︰数学系
考试日期(年月日)︰ 97.6.18
考试时限(分钟):120分钟
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
∞
1.Let X be a non-negative random variable. Shoe that E[X] = ∫( 1-F(x) ) dx,
0
where F(x) is the distribution function of X.
2.State the strong law of large numbers and the central law limit theorem.
3.Let X_1,...,X_n be a random sample from a normal distribution with mean μ
and variance σ^2. Find the mean and variance of n _
S^2 = Σ(X_i-X)^2 / (n-1),
i=1
_
where X is the sample mean of X_1,...,X_n.
4.Let X_1,...,X_n be a random sample from a geometric distribution
P(X = x) = p*(1-p)^(x-1)*I_{1,2,…}(x), and p have a
uniform prior distribution on [0,1].
(a)Derive the posterior distribution of p.
(b)Find the Bayes estimator of p based on the loss function
L( p,δ(X_1,...,X_n) ) =( δ(X_1,...,X_n)-p )^2.
5.Let X_1,...,X_n be a random sample from a uniform distribution on
[θ_l,θ_u].
(a)Find the maximum likelihood estimators of θ_l and θ_u.
(b)What is the joint distribution of the maximum likelihood estimators?
6.Let X_1,...,X_n be a random sample from a exponential distribution with the
density function f(x∣τ) = τ^(-1)*exp(-τ^(-1)*x)*I_{(0,∞)}(x).
(a)Find the maximum likelihood estimators of τ.
(b)Derive the sampling distribution of the maximum likelihood estimator of τ
(c)Find the uniform minimum variance unbiased estimator of τ.
7.Let X_1,...,X_n be a random sample from a density function
f(x∣θ)=θ*exp(-θx)*I_{(0,∞)}(x).
(a)Show that the rejection region of a likelihood ratio test of
H_O: θ = θ_0 versus H_A:θ≠θ_0 is of the form
_ _ _
{(X_1,...,X_n): X*exp(-θ_0*X) ≦ C }, where X is the sample mean of
X_1,...,X_n.
(b)Find a valid p-value for the above hypothesis.
8.Suppose that X_1,...,X_m are independent with X_i~Binomial(n,p_i).
(a)Derive a likelihood ratio test for the hypothesis
H_O:p_1 = p_2 = … = p_m versus the alternative hypothesis
H_A:p_i≠p_j for some i≠j.
(b)What is the large sample distribution of the test statistic?
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