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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