Let X1,X2,,,Xn be jointly noemal with EXi=0,EXi^2=1 for all i and cov(Xi,Xj)=ρ, i,j=1,2,,,(i≠j). What is the limiting distribution of n^-1Sn. Where Sn=∑Xk?
Let X1,X2,,,Xn be jointly noemal with EXi=0,EXi^2=1 for all i and cov(Xi,Xj)=ρ, i,j=1,2,,,(i≠j). What is the...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo 5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
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)
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n 9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
2. Suppose that X1, X2, .. , Xn are iid N(0, 02). Where i and o both assumed to be unknown. Let 0 = (i,a). Find jointly sufficient statistics for 0
8. Let X, X2, , xn all be be distributed Normal(μ, σ2). Let X1, X2, , xn be mu- tually independent. a) Find the distribution of U-Σǐ! Xi for positive integer m < n b) Find the distribution of Z2 where Z = M Hint: Can the solution from problem #2 be applied here for specific values of a and b?
Let X1, X2,· · ·iid B(1, x), i.e,P(X1= 1) =x= 1−P(X1= 0), where x∈ [0,1]. Let Sn = X1+X2+· · ·+Xn. What can you say about the limiting behaviour of Sn/n from strong law large number
Let Xi be iid with E(Xi) = 0 and Var(Xi) = 1 and let Sn = X1 + … + Xn. Consider the limiting behaviors of Sn/n and of Sn /n. Does either of these correspond to the LLN? to the CLT? Demonstrate using UNIF(–3, 3).
6. Let X1,..., Xn be a random sample from Uniform (0, 1). a) Find the exact distribution of U = – log(X(1)) where X(1) = min(X1, X2,..., Xn). b) Find the limiting distribution of n(1 – X(n)), where X(n) = max(X1, X2, ..., Xn).
Let X1...Xn be observations such that E(Xi)=u, Var(Xi)=02, and li – j] = 1 Cov(Xị,X;) = {pos, li - j| > 1. Let X and S2 be the sample mean and variance, respectively. a. Show that X is a consistent estimator for u. b. Is S2 unbiased for 02? Justify. - c. Show that S2 is asymptotically unbiased for 02.
4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let X(i) < X(2) < < X(n) be their order statistics. Define Yǐ = nX(1) and Ya = (n +1 - k)(Xh) Xk-n) for 1 < k Sn. Find the joint probability density function of y, . . . , h. Are they independent? 15In