课程名称︰高等统计推论一
课程性质︰数学系选修
课程教师︰江金仓
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2015/11/2
考试时限（分钟）：11:20~12:10
试题 :
1. (15%) Show that the axioms of finite additivity and continuity implies the
axiom of countable additivity.
∞
2. (15%) Let A ⊂...⊂A ⊂... and A = ∪ A . Show that lim P(A ) = P(A) via
1 n n=1 n n→∞ n
using the Komogorov axioms.
3. (15%) Let X be a random variable with the cumulative distribution function
F(x). Show that P(F(X) > y)≧(1-y) for y∈(0,1).
4. (15%) Let X have a Binomial(n,p), 0 < p < 1. Express the probability
P(X≦x), x=0,1,...,(n-1), in terms of the cumulative distribution of a
negative binomial random variable.
5. (5%)(10%) Let X be a negative binomial random variable with parameters r
and p. State and show the conditions so that X will converge to a Poisson
distribution with parameter λ.
6. (5%)(10%) State and show the conditions so that a hypergeometric
distribution can be approximated via a Poisson distribution.
7. (10%) Let X be a random variable with a probability density function
xθ－b(θ)
f(x|θ,ψ) = exp(——————＋c(x,ψ))
a(ψ)
for some specific functions a(‧),b(‧), and c(‧). Compute E[X|θ,ψ]
and Var(X|θ,ψ).