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）︰2010.04.19 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. (10%) Let X_1,...,X_n be a random sample from a distribution with mean μ and variance σ^2. Derive the mean of the sample variance n _ _ n (S_n)^2 = (n-1)^(-1) Σ (X_i - X_n)^2 , where X_n = n^(-1)Σ X_i . i=1 i=1 2. (10%) Let X follow a normal distribution with mean μ and variance σ^2. Show that μ_2n = (2n)!σ^2n / (2^n)n! and μ_2n+1 = 0, n = 1, 2,.... p 3.(10%) Let Y = Σ X_i with X_1,...,X_p being independent exponential random i=1 variables 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. (4a) (7%) Compute the moment generating function of X. (4b) (8%) Assume that n→∞, p→0, and 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. (10%) Letρ be athe correlation coefficient of X and Y. Show that ｜ρ∣≦ 1 if and only if P(Y = aX+b) = 1 for a≠0. 6. (10%) Let the distribution of U conditioning on T = t be Uniform(0,t) and T follow an exponential distribution with rate λ > 0. Compute E[U] and Var(U) 7. (10%) Show that Var(Y) = E[Var(Y｜X)] + Var(E[Y｜X]). 8. (10%) Let X be an exponential distribution with standard deviation σ. Find P(｜X-E[X]｜ > κσ) and compare it to the upper bound from the Chebyshev's inequality. 9. (10%) Let X_1,...,X_n be a random sample from Uniform(0,1). Find the mean of (X_(n) - X_(1)), where X_(1) and X_(n) are the minimum and maximum order statistics, respectively. 10. (10%) Let (X,Y) be a random point uniformly distributed on a unit disk. Show that Coυ(X,Y) = 0, but that X and Y are not independent. --

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