課程名稱︰統計學
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）︰97.4.23
考試時限（分鐘）：120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.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
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※ 編輯: fatsheepgary 來自: 140.112.243.231 (07/06 17:33)