18. Supppse(Xy, n 1) are iid with E(%)--0, and E(X )-1. Set S,- Li i-1 Xi. Show Sn →0 n1/2 log n almost surely. 18. Supppse(Xy, n 1) are iid with E(%)--0, and E(X )-1. Set S,- Li i-1 Xi. Show Sn →0 n1/2 log n almost surely.
Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a MVU estimator for m. Problem 1: Let (Xi,..., Xn) denote a random variable from X having a Log-normal density fx (x) = d(L m)/ x, x 〉 0 n(x) - where m is an unknown parameter. Show n-1 Σ'al Ln(X) is a...
1. A sequence of random variables Xn satisfy Xn _>X in probability and E(Xn) -> E(X) for some random variable X (a) Show that E([X, - X|) -> 0 if Xn >0 for all n (b) Find a counterexample satisfying E(X,n - X) A0 if X are not non-negative. 1. A sequence of random variables Xn satisfy Xn _>X in probability and E(Xn) -> E(X) for some random variable X (a) Show that E([X, - X|) -> 0 if Xn...
Let X be distributed as N(0, 1). Define Xn (1)"X, n 1,2, a. [3 pts] Show that Xn-X. b. [3 pts] Show that Xn -» X
3.10 (i) If X1, , Xn are i.i.d. according to the exponential density e-", r >0, show that (2.9.3) P [X(n)-log n < y]- e-e-v, -00 < y < oo. (ii) Show that the right side of (2.9.3) is a cumulative distribution function. (The distribution with this edf is called the ertreme value distribution.) (iii) Graph the cdf of X(n)-log n for n = 1, 2, 5 together with the mit e-e" (iv) Graph the densities corresponding to the cdf's...
7. Show that σ2 E(X-0 and Var(X if X1, . . . , Xn are independent and identically distributed with E(Xi) = 0 and E(X2) = σ2 for i = 1,-.. , n
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...
Problem 5: Evaluate log(x) Jo 4+2 0 3. Show that 2x cos(e) Jo 1-cos(0) Problem 5: Evaluate log(x) Jo 4+2 0 3. Show that 2x cos(e) Jo 1-cos(0)
Equation(1): 2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E(TIXo = 2) using a geometric distribution. (ii) Use (1) to compute E(T Xo 2). 0), and Vi(n) i rst-step analysis to show that 2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E(TIXo = 2) using a geometric distribution. (ii) Use...
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.