need help with (b) 4. Consider the random walk on the state space 10,1,2, ...^, with transition probabilities for i - ,4,. . ., given by if j=1+1 otherwise, 0 an Pol 1 . As usual, p+q 1, and we assum...
Consider the Markov chain with state space S = {0,1,2,...} and transition probabilities I p, j=i+1 pſi,j) = { q, j=0 10, otherwise where p,q> 0 and p+q = 1.1 This example was discussed in class a few lectures ago; it counts the lengths of runs of heads in a sequence of independent coin tosses. 1) Show that the chain is irreducible.2 2) Find P.(To =n) for n=1,2,...3 What is the name of this distribution? 3) Is the chain recurrent?...
Consider a Markov chain with transition probabilities p(x, y), with state space S = {1, 2, . . . , 10}, and assume X0 = 3. Express the conditional probability P3(X6 =7, X5 =3 | X4 =1, X9 =3) entirely in terms of (if necessary, multi-step) transition probabilities.
Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning Consider the process E here Xn is the outcome of a die on the nth roll at XnEN is a Markov chain. (b) Determine the state space S and the transition matrix P (with, as usual, reasoning
Consider a Markov chain with state space S = {1, 2, 3, 4} and transition matrix P= where (a) Draw a directed graph that represents the transition matrix for this Markov chain. (b) Compute the following probabilities: P(starting from state 1, the process reaches state 3 in exactly three time steps); P(starting from state 1, the process reaches state 3 in exactly four time steps); P(starting from state 1, the process reaches states higher than state 1 in exactly two...
Consider a two state Markov chain with one-step transition matrix on the states 1,21, , 0<p+q<2. 91-9 ' Show, by induction or otherwise, that the n-step transition matrix is Ptg -99 Based upon the above equation, what is lim-x P(Xn-2K-1). How about limn→x P(Xn-
2. Problem 2.5. Consider a random walk on 10..... which movies left and right with respective probabilities a and p. If the walk is at 0 it transitions to 1 on the next step. If the walk is at k it transitions to k-1 on the next step. This is called random walk with reflecting boundaries. Assume that k 3, =1/4, p = 3/4, and the initial distribution is uniform. For the following, use technology if needed. (a) (10.1.X2 }...
P is the (one-step) transition probability matrix of a Markov chain with state space {0, 1, 2, 3, 4 0.5 0.0 0.5 0.0 0.0 0.25 0.5 0.25 0.0 0.0 P=10.5 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.0 0.5 0.5/ (a) Draw a transition diagram. (b) Suppose the chain starts at time 0 in state 2. That is, Xo 2. Find E Xi (c)Suppose the chain starts at time 0 in any of the states with...
Problem 3.3 (10 points) Consider a two-state continuous time Markov chain with state space 11,2) and transition function (a) Find P(X-21 Xo = 1]. (b) Find P[X5 1, X2 2 X1-1] Problem 3.3 (10 points) Consider a two-state continuous time Markov chain with state space 11,2) and transition function (a) Find P(X-21 Xo = 1]. (b) Find P[X5 1, X2 2 X1-1]
Q4 and Q5 thanks! 4. Consider the Markov chain on S (1,2,3,4,5] running according to the transition probability matrix 1/3 1/3 0 1/3 0 0 1/2 0 0 1/2 P=10 0 1/43/40 0 0 1/2 1/2 0 0 1/2 0 0 1/2 (a) Find inn p k for j, k#1, 2, ,5 (b) If the chain starts in state 1, what is the expected number of times the chain -+00 spends in state 1? (including the starting point). (c) If...
Question 5 9 marks Consider a Markov chain {YTheN with state space S = {1,2,3,4), initial distribution Po (0.25,0.25, 0.5,0), and transition matrix 1/3 2/3 0 0 p 1/6 1/2 1/30 0 4/9 4/9 1/9 0 0 5/6 1/6 2(a) Find the equilibrium probability distribution T (b) Find the probability P(-1%-3. Ya-1). Question 5 9 marks Consider a Markov chain {YTheN with state space S = {1,2,3,4), initial distribution Po (0.25,0.25, 0.5,0), and transition matrix 1/3 2/3 0 0 p...