[試題] 98下 江金倉 統計學 期中考
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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