Let X1, X2, . . . , n be iid random variables with common CDF F. Generate the random variable defined by X = min 1<i<n (Xi) in terms of the inverse of F.
1 [3]. Let X1,X2, X3 be iid random variables with the common mean --1 2-4 and variance σ Find (a) E (2X1 - 3X2 + 4X3); (b) Var(2X1 -4X2); (c) Cov(Xi - X2, X1 +2X2).
Let X1, X2, ... be independent continuous random variables with a common distribution function F and density f. For k > 1, let Nk = min{n>k: Xn = kth largest of X1, ... , Xn} (a) Show Pr(Nx = n) = min-1),n>k. (b) Argue that fxx, (a) = f(x)+(a)k-( ++2)(F(x)* (c) Prove the following identity: al= (+*+ 2) (1 – a)', a € (0,1), # 22. i
3. Let X1, X2, . . . , Xn be random variables with a common mean μ. Sup- pose that cov[Xi, xj] = 0 for all i and A such that j > i+1. If 仁1 and 6 VECTORS OF RANDOM VARIABLES prove that = var X n(n- 3)
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X be a random variable with P(X = 0) = 1. (a) what is the CDF of Xn? (b) Does Xn converge to X in distribution? in probability?
4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ?
4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ?
Let X1,X2,...,Xn be iid N(μ,1) random variables. Find the MVUE of θ=μ2.
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
2. Let X1, X2, X3 ..., X, be iid b(1, p) random variables. Let Sn = 27-1Xthen prove that Sn-E(Sn) N(0,1) as n +00. (Sn)
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...