Let X be distributed as N(0, 1). Define Xn (1)"X, n 1,2, a. [3 pts] Show...
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
3. Let U1, U2,. be a sequence of independent Ber(p) random variables. Define Xo 0 and Xn+1-Xn +2Un-1, 1,2,.. (a) Show that X, n 0,1,2, is a Markov chain, and give its transition graph. (b) Find EX and Var(X) c)Give P(X
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
5. Let X1, X2, . . . , Xn be independently distributed as N(u, σ2). Define 71 -1 /Users/rumi3/Downloads/Linear-Regression-Analysis-Seber pdf MOMENT GENERATING FUNCTIONS AND INDEPENDENCE 13 and (a) Prove that var[S2-2σ4/(n-1). (b) Show that Q is an unbiased estimate of σ2. (c) Find the variance of Qand hence show that as n → oo, the effi- ciency of Q relative to S2 is
5. Let X1, X2, . .. , Xn be independently distributed as N(μ, σ2). Define 7t n-1 ー1 Users/rumi3/Downloads/Linear-Regression-Analysis-Seber.pdf MOMENT GENERATING FUNCTIONS AND INDEPENDENCE 13 and n-1 2(n-1 i=1 (a) Prove that var[S2-2c4/(n-1). b) Show that Q is an unbiased estimate ofa (c) Find the variance of Q and hence show that as n → oo, the effi- ciency of Q relative to S2 is
4. Fix > 0. For n > λ let Xn be Geometric(A/n). Show that X n/n converges in distribution to an Exponential(A). (Hint: again, compute moment generating functions.)
& Let Yn = ao Xn ta, x n- do xn + a, Xn-1 ; n =1,2,... where Xi are iid RVs u ; n = 1, 2, with equal moon o and va siance 2.1 95 { yn in 21% SSS? Is {Yn: n 21 } WSS?
2. Let {xn}nEN be a sequence in R converging to x 0. Show that the sequence R. Assume that x 0 and for each n є N, xn converges to 1. 3. Let A C R". Say that x E Rn is a limit point of A if every open ball around x contains a point y x such that y E A. Let K c Rn be a set such that every infinite subset of K has a limit...
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...
3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2 < oo. Find cov(X,-x,x) for i 1,..,n. 3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2