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density function
创建日期：2024-07-17 17:51:51
所属分类：数学mathematics
标签： function
density function

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Q1 (30 points)

Suppose the cdf of (X1 , X2) is given by

F(x1,x2) =1-e-x1 + e-x2 + e-x1-x2

for x1 ≥ 0,x2 ≥ 0 and is 0 otherwise.

a）(5 marks) Determine the joint probability density function of (X1 , X2).

b）(5 marks) Are  and  statistically independent? Justify your answer.

c）(5 marks) Determine the mean vector and variance matrix of (X1 , X2).

d）(5marks) Determine the mean vector and variance matrix of where

 

 

e）(5 marks) Are  and  in (d) statistically independent? Justify your answer.

f）(5 marks) Determine the joint density function of (Y1 , Y2)

 

Q2 (20 points) 

Suppose Z0 , Z1,…are i.i.d.N(0,1). With T = {1,2,…}, define the process by {(t，Xt) : t∈T }  by Xt = ZtZt-1

a) (5 marks) Determine the mean and autocovariance functions of the  Xt process.

b) (5marks)If Z0=1 , Z1=3. , Z2=-4, Z3=2 then plot the first three values of the sample function of the Xt  process.

(c) (10 marks) Determine the moment generating function (mgf) mXt(s) of Xt (Hint: use the theorem of total expectation.) Does the mgf exist for all s ∈R1

 

Q3 (20 points)  

Suppose that h : R4 → R1 is given by

 

 

(a) (5 marks) Prove that is h convex.

(b) (5 marks) Suppose that X = (X1, X2, X3, X4)’ has a joint distribution with mean vector and variance matrix given by


 

Determine a general lower bound on E(h(X)) .

(c) (5marks) If X～N4(μ,∑) , then determine E(h(X)) exactly.

(d) (5 marks) What is the best afinepredictor of  when X2 =(X3, X4) when X1 =  (X1, X2)=(x1,x2) is observed? Under what conditions is this also the best predictor and explain what ”best”

 

Q5 (10 points)

(10 marks) Suppose that Z0,Z1,… are i.i.d N(0,1) and Xn = 1/n + αZn + βZn-1 . Determine whether or not {(n,Xn ):n∈ N } is a stationary Gaussian process.

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