作者fatsheepgary (老羊)
看板NTU-Exam
标题[试题]96下 江金仓 统计学 期中考
时间Sun Jul 6 17:32:47 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)︰97.4.23
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.Define or explain the following terms or statements.
(a)Baye's rule. (b)strong law of large numbers. (c)converges in distribution.
(d)random sample. (e)statistic.
2.Let Y=exp(Z) and Z be a normal random variable with mean μ and variance
σ^2. Compute the mean and variance of Y.
3.Let p with X_1,...,X_p being independent exponential random variables
Y= Σ X_i
i=1
with a common parameter λ. Derive the distribution of Y.
4.Let X be a binomial random variable with parameters n and 0 < p < 1.
(a) Compute the moment generating function of X.
(b) Assume that n→∞, p→0, λ_n = np →λ with λ > 0. Show that the moment
generating function converges to the moment generating function of a
Poisson random variable with parameter λ.
5.Let X and Y be mutually independent Chi-square random variables with the
degrees of freedom m and n. Derive the probability density function of
U=nX/mY and compute the mean of U for n > 2.
6.Let X_1,...,X_n be a random sample from a uniform distribution U(0,1), V be
a independent U(0,1) random variable, and X_(k) and X_(m) be the kth and
mth order statistics of {X_1,...,X_n}, 1 < k < m <n.
Compute the probability P( X_(k) < V <X_(m) ).
7.Let ρ be the correlation coefficient of X and Y. Show that ∣ρ∣≦ 1
and ∣ρ∣= 1 if and only if P(Y = aX + b) = 1 for a≠0.
8.Show that E[Var(Y∣X)] ≦ Var(Y).
9.Let X_1,...,X_n be a random sample from a distribution with mean μ and
variance σ^2.
(a)Derive the mean of the sample variance n _
(S_n)^2 = 1/(n-1)*Σ(X_i-X_n)^2
_ n i=1
where X_n = 1/n*Σ X_i.
i=1 _
(b)Show that the sample mean X_n converges in probability to μ.
10.Let X_1,...,X_n be a random sample from a normal distribution with mean μ
and variance σ^2.
(a)Show that
(n-1)*S_n^2 2
——————— ~ χ
σ^2 n-1.
_
(b)Show that √n *(X_n-μ)
———————— ~ t_n-1.
S_n
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.112.243.231
※ 编辑: fatsheepgary 来自: 140.112.243.231 (07/06 17:33)