科目：高等統計學一
教授：銀慶剛 教授
試別：期末考
時間：take home exam
by ajarist
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1.Let X be any RV, and suppose that the MGF of X, M(t)=E{exp(tx)},
exists for every t>0. Then for any t>0,
P{tX > s^2+logM(t)} < exp(-s^2)
2.Let (X1,X2,X3) be an RV with joint PMF
f(x1,x2,x3)=1/4 ╭ if(x1,x2,x3) 屬於 A
=0 ╰ o.w.
where A= {(1,0,0),(0,1,0),(0,0.1),(1,1,1)}
Are X1,X2,X3 independent? Are X1,X2,X3 pairwisw independent?
Are X1+X2 and X3 independent?
3.Let X,Y be iid RVs with commom PDF
f(x)= exp(-x) ╭ if x>0
= 0 ╰ if x≦0
Show the U=X+Y and Z=X/(X+Y) are independent.
4.Let X1,X2 be iid with common Poisson PMF
P(Xi=x) = exp(λ) * λ^x /x! , x=0,1,2,..., i=1,2,
where λ>0 is a constant.
Let X(2) = max (X1,X2) and X(1) = min (X1,X2).
Find the PMF of X(2)
5.If X is a nondegenerate RV with finite expectation and such that X≧a>0
then E {√(X^2-a^2)} < √{(EX)^2-a^2}
6.Let X have PMF P(X=x) = exp(λ) * λ^x /x! , x=0,1,2,...
and suppose that λ is a realization of a RV Λ with PDF
f(λ)= exp(-λ) , λ>0
Find E{exp(-Λ) | X=1}.
7.If X1,X2,...Xn are independent RVs with Xi~Poisson(λi), i=1,2...n,
the conditional distribution of X1,X2,...Xn,
given Σ(Xi)=t,is multinomial with parameters t,
λ1/Σλi,...,λn/Σλi.
8.X~Gamma(a,b) and let Y~U(0,X).
(a). Find the PDF of Y.
(b). Find the conditional PDF of X given Y=y.
(c). Find P(X+Y≦2).
9.Let X1,X2...be iid RVs with mean 0,variance 1, and E(Xi^4)<∞
Find the limiting distribution of
X1*X2+X3*X4+...+X2n-1*X2n
Zn = √n * ---------------------------
X1^2+X2^2+...+X2n^2
10.Let X and Y be independent and identically distributed N(0,σ^2) RVs.
Show the X^2 + Y^2 and X / √(X^2+Y^2) are independent.
11.Let Sn have a Chi-square distribution with df=n.
Show that the limiting distribution of √Sn-√n is N(0,1/2).
This is known as Fisher's approximation.
12.Let X and Y be two random variables.
What assumptions can make X and Y independent? List as many as you know.
∮完卷∮
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