課程名稱︰高等統計推論一
課程性質︰數學系選修
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）︰2015/10/22
考試時限（分鐘）：16:30~17:20
試題 :
1. (15%) Let (Ω,A,P) be the probability space, A ,..., A ,...∈A with
1 n
∞
P(A ) = 1 for all i. Show that P(∩ A ) = 1.
i i=1 i
2. (10%) Let X ,...,X ,... be random variables. Show that inf X is a random
1 n n n
variable.
3. (7%)(8%) State and prove the second Borel-Cantelli lemma.
4. (15%) Find an example in which two random variables have different
distributions but the same moments.
5. (15%) Find an example in which the moment generating function does not
exist but the moments exist.
2
6. (15%) Suppose that E[X ] < ∞ and X = min{X,C} for a constant C. Show that
C
Var(X ) ≦ Var(X).
C
7. (7%)(8%) Let X have a probability density function
p
f(x|θ) = h(x)c(θ)exp(Σθ t (x)) with the natural parameter space
j=1 j j
∞ p
H = {θ:∫ h(x)exp(Σθ t (x)) dx < ∞}. Derive the expectation and
-∞ j=1 j j
p
variance of Σ t (X).
i=1 i
E[e^(tΣt_i(X))]=c(θ)/c(θ+t)∫h(x)c(θ+t)exp(Σ(θ+t)t_i(x))dx