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 ---------------------------------------------------------------------- 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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