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-time steps).
(c) If the process starts from state 3, provide the states which are not attainable in exactly two-time steps.
Consider a Markov chain with state space S = {1,2,3,4} and transition matrix P = where...
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 the Markov chain X0,X1,X2,... on the state space S = {0,1} with transition matrix P= (a) Show that the process defined by the pair Zn := (Xn−1,Xn), n ≥ 1, is a Markov chain on the state space consisting of four (pair) states: (0,0),(0,1),(1,0),(1,1). (b) Determine the transition probability matrix for the process Zn, n ≥ 1.
2. Consider a Markov chain with state space S 1,2,3,4) with transition matrix 1/3 2/3 0 0 3/4 1/4 00 0 0 1/5 4/5 0 0 2/3 1/3, (a) (10 points) Is the Markov chain irreducible? Explain your answer ive three examples of stationary distributions.
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 a Markov chain with state space S = {0, 1, 2, 3} and transition probability matrix P= (a) Starting from state 1, determine the mean time that the process spends in each transient state 1 and 2, separately, prior to absorption. (b) Determine the mean time to absorption starting from state 1. (c) Starting from state 1, determine the probability for the process to be absorbed in state 0. Which state is it then more likely for the process...
Problem 7.4 (10 points) A Markov chain Xo, X1, X2,.. with state space S = {1,2,3,4} has the following transition graph 0.5 0.5 0.5 0.5 0.5 0.5 2 0.5 0.5 (a) Provide the transition matrix for the Markov chain (b) Determine all recurrent and all transient states (c) Determine all communication classes. Is the Markov chain irreducible? (d) Find the stationary distribution (e) Can you say something about the limiting distribution of this Markov chain? Problem 7.4 (10 points) A...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
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...
Markov Chains Consider the Markov chain with transition matrix P = [ 0 1 1 0]. 1) Compute several powers of P by hand. What do you notice? 2) Argue that a Markov chain with P as its transition matrix cannot stabilize unless both initial probabilities are 1/2.