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）︰ 97.6.18 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.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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