8. A particle moves on a circle through points that have been marked 0, 1, 2, 3, 4 (in a clockwise order). Theparticle starts at point 0. At each step it has probability 0.3 of moving one point clockwise (0 follows 4) and 0.7of moving one point counterclockwise. Let Xn (n ≥ 0) denote its location on the circle after step n. {X₂} is aMarkov chain.(a) Construct the (one-step) transition matrix.(b) Find the n-step transition matrix P(n) for n= 40, 80, and 81. Does this Markov chain have steady-state?

a) The particle can move to any of the five points on the circle with probability 1 in one step. Let…

Answered: 8. A particle moves on a circle through…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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4. Let (X₂)o be a Markov chain with state space S = {1,2,3} and transition matrix .8 .2 O .5 .5 0 0 0 1 Find two different stationary distributions for this Markov chain. P =
3. A discrete Markov model has state space equal to E {0, 1, 2}. The transition probabilities p are independent of r and recorded in the matrix P defined below. .7 .2 .1 P = .1 .6 .3 1 Assume that Lynn is in state 0 at time 0, and that i = .04. (a.) Determine the probability that Lynn is in state 0 at time 3. (b.) Determine the probability that Lynn transitions from state 1 at time 3 to state 2 at time 4.
2. Consider a Markov chain (Xn) with state space S = {1, 2, 3, 4, 5} and transition matrix 0.5 0.5 0 0 0 0.4 0.6 0 0 0 P = 0 0.3 0.3 0.4 0 0 0 0 0.6 0.4 000 0.6 0.4 One stationary distribution for this Markov chain is (4, 5, 0, 0, 0). Suppose the Markov chain (X) is started from the initial distribution λ = (0.1 0 0.7 0.2 0). What are the limiting probabilities limn→∞ P(X = i) for each i € S?
5. Suppose {Xn, n 2 0} is a Markov chain with state space (0, 1,2} and transition probability matrix 0.6 0.2 0.2 P= 0.8 0.2 0.5 0.5 Find the classes for this Markov chain and classify them as transient or recurrent.
I need question 4 only
Consider the Markov chain with state space S = {0, 1, 2,...} and transition probabili- ties Poo = 1 - P. P02 = P, and Pr,x+2= P₁ P2,2-1=1-p, for x = 1,2,... For which values of p is this a transient chain?
Find the limiting distribution for this Markov chain. Then give an interpretation of what the first entry of the distribution you found tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology.
Please answer #28 and explain your reasoning.
Consider a Markov chain with two possible states, S = {0, 1}. In particular, suppose that the transition matrix is given by Show that pn = 1 P = [¹3 В -x [/³² x] + B x +B[B x] x 1- B] (1-x-B)¹ x +B x -|- -B ·X₁ В
Consider a random walk on the graph with 6 vertices below. Suppose that 0 and 5 are absorbing vertices, but the random walker is more strongly attracted to 5 than to 0. So for each turn, the probability that the walker moves right is 0.7, while the probability he moves left is only 0.3. (a) Write the transition matrix P for this Markov process. (b) Find POO. (c) What is the probability that a walker starting at vertex 1 is absorbed by vertex 0? (d) What is the probability that a walker starting at vertex 1 is absorbed by vertex 5? (e) What is the probability that a walker starting at vertex 4 is absorbed by vertex 5? (f) What is the probability that a walker starting at vertex 3 is absorbed by vertex 5? What is the expected number of times that a walker starting at vertex 1 will visit vertex 2? (h) What is the expected number of times that a walker starting at vertex 1 will visit vertex 4? N;B The diagram is a horizontal line showing points 0, 1, 2, 3, 4, and 5
1. A Markov chain has state space {0, 1, 2} and transition probability matrix .1 .5 .4 P = 0 0 1 .2 .3 .5 (a) the probability of going from state 0 to state 2 in 2 transitions (b) the steady-state distribution, if it exists Determine:
A path of length k in a Markov chain {X,,n= 0,1,...} is a sequence of states visited from step n to step n+k. Let pij be the transition probability from state i to state j. Show that starting from state i,, the probability that the chain follows a particular path i, - i„.1 - in2 - ... - i,t is given by %3D "x*\":= "X)d = i2.,Xpsk = inrk | X, = i,) n+1 n+l> n+2 n+k = P Pu.

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