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）︰2016/01/04
考试时限（分钟）：11:20~12:10
试题 :
1. (10%) Let (f (t),f (t)) and (S (t),S (t)) be the probability density
T C T C
functions and the survivor functions of continuous random variables (T,C)
with support (0,∞). Derive the joint p.d.f of X = min{T,C} and δ=I(X = T)
under the independence of T and C.
δ 1-δ
f (x,δ)=[S (x)f (x)] [S (x)f (x)]
X,δ C T T C
2. Let U ,...,U ,... be independent random variables from Uniform(0,1) and X
1 n
have probability density function P(X=x) = (1/[(e-1)x!])I (x).
{1,2,3,...}
(2a) (8%) Find the distribution of Z = min{U ,...,U }.
1 X
k
(2b)(8%) Define N = min{k:S >1} with S = Σ U .Compute the expectation of N.
k k i=1 i
3. (10%) Let X ,...,X be a random sample from the probability density function
1 n
a-1 a
f (x) = (ax /θ )I (x) and X <...< X stand for the corresponding
X {(0,θ)} (1) (n)
order statistics. Find the joint distribution of (X /X , X /X ,...
(1) (2) (2) (3)
T
, X /X ) .
(n-1) (n)
4. (7%)(7%) Give examples to show that convergence in probability cannot imply
1
the convergence almost surely and convergence in L .
T
5. (10%) Let (X ,...,X ) have a multinomial distribution with n trials and the
1 m
cell probabilities p ,...,p . Derive the probability distribution of X
1 m i
conditioning on X = x for i≠j.
j j
6. (3%)(7%) State and show the Holder's inequality.
T T T
7. (10%) Let X = (X ,X ) be a p-variate normal distribution with mean vector
1 2
μ and variance-covariance matrix Σ. Suppose that Σ is positive definite.
Derive the distribution of X conditioning on X = x .
1 2 2
8. (5%)(5%) Let X ,...,X be a random sample with a continuous distribution
1 n
function F(X) and X ,...,X be the corresponding order statistics.
(1) (n)
Compute the mean and variance of F(X ), i=1,...,n.
(i)
Note that F(X ) ~ U , where U ,...,U ~Uniform(0,1).
(i) (i) 1 n
9. (10%) Let X ,...,X be a random sample from a normal distribution with mean
1 n
2 _ n 2 n _ 2
μ and variance σ , X = Σ X /n, and S = Σ (X - X ) /(n-1). Find a
n i=1 i n i=1 i n
2 2 2 p
function g(‧) of S such that E[g(S )|(μ,σ )] = σ for (n + p) > 1.
n n
2 p/2
Hint: Try E[(S ) ].
n