Help please! Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic...
Help please!
Let {Zn}n=0 be iid. Bernoulli random, vari- + Zn. ables with PZ-0] = p and PZ-1] 1-p. Define Sn-Zo + Which of the following processes is a Markov chain? 1. An S For each process that is a Markov chain, find its transition matriz. For each process Xn E An, Bn, Cn, Dn that is not a Markov chain, find a pair of states i and j so that P[Xn+ 1 = ilXn-j, Xn-1-k] depends on k.
Problem! (20p). Let E be a countable set, (F, F) an event space, f : E × F ? E a random variable, and (Un)1 a sequence of i.i.d. random variables with values in F. Set Xo r for some xe E, and for n e Z let Xn f(Xn, Unti). Show that (X)n is a Markov chain and determine its transition matrix
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
Stochastic Processes Markov
5 Let Xn, n 0, be the two-state Markov chain. (a) Find Po(To - n). (b) Find Po(T n).
Let Z1, Z2, . . . be a sequence of independent standard normal random variables. Define X0 = 0 and Xn+1 = (nXn + (Zn+1))/ (n + 1) , n = 0, 1, 2, . . . . The stochastic process {Xn, n = 0, 1, 2, } is a Markov chain, but with a continuous state space. (a) Find E(Xn) and Var(Xn). (b) Give probability distribution of Xn. (c) Find limn→∞ P(Xn > epsilon) for any epsilon > 0.
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
2 is the question
Question 4 [35 marks in total] An n xn matrix A is called a stochastic matriz if it satisfies two conditions: (i) all entries of A are non-negative; and (ii) the sum of entries in each column is one. If the (i, j) entry of A is denoted by aij for i,j e {1, 2, ..., n}, then A is a stochastic matrix when aij > 0 for all i and j and in dij =...
Consider a process {Xn, n = 0,1, ... }, which takes on the values 0,1, or 2. Suppose P{Xn+1 = ||Xn = i, Xn-1 = in-1,..., X0 = i +0} = when n is even when n is odd where - P = - Phl = 1, i = 0,1,2. Is {Xn, n = 0,1,... } a time-homogeneous Markov chain? If not, then show how, by enlarging the state space, we may transform it into a time- homogeneous Markov chain.
1. Let X1, ·s, Xn be independent random variables taking values 0 or 1 withP(Xi=1)=eθ-ai /(1+eθ-ai ), i=1, ……, nfor some given constants ai. Find a one-dimensional sufficient statistic for θ.