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）︰2015/11/12
考試時限（分鐘）：15:30~17:20
試題 :
1. (10%) Let (Ω,A,P) denote the probability space and A ,...,A ,...∈A with
1 n
∞ ∞ ∞
Σ P(A ) < ∞. Compute the probability P( ∩ ∪ A ).
n=1 n k=1 n=k n
2. (10%) Let X be a random variable with Var(X) = 0. Compute the probability
P(X = E[X]).
3. (10%) Suppose that X is a nonnegative random variable with finite mean.
∞ ∞
Show that Σ P(X > k)≦E[X]≦ Σ P(X > k).
k=1 k=0
4. (10%) Let Φ (t) be the characteristic function of X and a and b be real-
X
valued constants with b > a. Show that
1 1 1 T exp(-ita)-exp(-itb)
P(a<X<b) + — P(X=a) + — P(X=b) = lim ——∫ ——————————Φ (t)dt.
2 2 T→∞ 2π -T it X
5. (10%) Let X ~ Gamma(n,1) and W = (X - E[X])/√(Var(X)). Derive the limiting
distribution of W.
6. (10%) Let Z be a standard normal random variable. Show that
2
z -z
P(|Z|≧z) ≧ √(2/π) ———— exp(——).
2 2
1 + z
2
7. (10%) Let X ~ Poisson(λ) and Y ~ χ . Express the cumulative distribution
p
function of X via the cumulative distribution function of Y.
8. (10%) Let X ~ NegativeBinomial(r,p),where 0<p<1 and r is a positive integer.
Derive the limiting distribution of Y = 2pX as p→0.
9. (10%) Assume that a family of p.d.f.s {f (t|θ):θ∈Θ} has monotone
T
likelihood ratio (non-decreasing case) in T. Show that P(T > t |θ)
0 2
≧ P(T > t |θ) for all θ > θ.
0 1 2 1
10.(10%) Suppose that the counting process N(t) follows a non-stationary
Poisson process with intensity function λ(t). Compute the probability
P(N(t) = m) for each nonnegative integer m.