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）︰2015/12/14
考試時限（分鐘）：11:20~12:10
試題 :
1. Let X be a p-variate elliptical distribution with mean vector μ and
T -1
non-singular variance-covariance matrix Σ, i.e. f (x)=g((x-μ) Σ (x-μ))
X
for some function g. Derive the characteristic function of X.
φ(t)=exp(it^T μ)g(t^T Σt)
2. (10%) Let C be a p×p constant matrix and X be a p×1 random vector with
mean vector μ and variance-covariance matrix Σ. Derive the expectation of
T
X CX.
3. (10%) Suppose that X , X , and X are mutually independent Poisson(λ)
1 2 3
random variables. In addition, let Y = X + X , Y = X + X , and Z = I(Y =0),
1 1 3 2 2 3 i i
i=1,2. Compute the correlation of Z and Z .
1 2
4. (10%) Let X and Y be mutually independent continuous random variables with
the corresponding cumulative distribution functions (density functions)
F (x) (f (x)) and F (y) (f (y)). Derive the distribution of Y conditioning
X X Y Y
on Z = 1, where Z = I(X = Y).
5. (10%) Suppose that random numbers of U(0,1) can be easily generated. State
2
one algorithm to generate a random variable of N(μ,σ ).
2
6. (10%) Let X be χ , i=1,...,n, and X ,...,X are mutually independent.
i υ 1 n
i
n
Derive the distribution of X / Σ X .
1 i=1 i
Note that the sum of chi-square is gamma. Then derive the distribution
of X/X+Y,where Y is the sum of some chi-square.
7. (10%)(5%)(5%) Let X|P = p ~ NegativeBinomial(r,p) and P ~ Beta(α,β).
Derive the marginal distribution of X and compute the expectation and
variance of X.
8. (10%) Let X~Poisson(θ) and Y~Poisson(λ) be independent, and Z = X + Y.
Derive the distribution function of X conditioning on Z = z.
T
9. (5%)(5%) Let (X,Y) be a bivariate random vector with finite variances.
Compute Cov(X,Y-E[Y|X]) and show that Var(Y-E[Y|X]) = E[Var(Y|X)].