作者fatsheepgary (老羊)
看板NTU-Exam
标题[试题] 97上 江金仓 高等统计推论 期中考
时间Tue Nov 11 19:57:27 2008
课程名称︰高等统计推论(上)
课程性质︰必修
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)︰97/11/11
考试时限(分钟):110分钟
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
1. Let (Ω,A*,P) denote the probability space and A_1,...,A_n belong to A.(注1)
n n
(1a) Show that P(∩A_i) = ΣP(A_i) - (n-1).
i=1 i=1
n
(1b) If P(A_i)=1 for all i, show that P(∩A_i) = 1.
i=1
(1c) Let B belong to A* with P(B)>0. Show that P_B(A)=P(A|B) for all A
belong to A* is a probability function.
∞∞ ∞
(1d) Show that P(∩∪A_n) = 0 if ΣP(A_n)<∞.
k=1n=k n=1
2. Let X be a random variable with Var(X) = 0. Show that P(X = E[X]) = 1.
3. Let X ~ Gamma(n,1), W = (X-E[X])/√(Var(X)), and Φ(z) be the cumulative
distribution function of a standard normal distribution.
(3a) Show that lim P(W≦z) = P(Z≦z) for all z.
n→∞
(3b) Show that((√n)/Gamma(n))*(z√n + n)^(n-1)*exp(-z√n - n) ≒
1/ (√2π)*exp(-z^2/2) and z=0 gives Stirling's formula.
4. Let N_t denote the number of events occuring within the time period [0,t]
and T be the time between two successive events.
(4a) Write the nessssary conditions so that N_t follows a Poisson
distribution with rate λ.
(4b) Derive the probability density functions of N_t and T.
5. Let X ~ Poisson(θ),Y ~ Poisson(λ) are independent random variables,
Z=X+Y. Find the distribution function of X condition on Z=z.(注2)
6. Let X_1,...,X_p be independent random variables and U_i = g_i(X_i) with g_i
being a measurable function, i=1,...,p. Show that U_i's are mutually
independent.
7. Let X_i ~ Gamma(α_i,β), i=1,...,m, be mutually independent. Derive the
m
distribution of W = ΣX_i.
i=1
8. Assume that a family of p.d.f.s {f_T(t|θ):θ belong to Θ} has monotone
likekihood ratio(non-decreasing case) in T. Show that
P(T >t_0|θ_2) ≧ P(T >t_0|θ_1) for all θ_2 > θ_1.
注1: A* 是sigma algebra的符号,不过这里好像打不出来.
注2: 这题是老师修改题目後写在黑板上的,所以可能会有点出入.
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