Suppose a Markov chain has state space S (0, 1,2) and states 0 and 1 are known to be transient. N...
Consider the Markov chain with state space {0, 1,2} and transition matrix(a) Suppose Xo-0. Find the probability that X2 = 2. (b) Find the stationary distribution of the Markov chain
A Markov chain {Xn, n ≥ 0} with state space S = {0, 1, 2, 3, 4,
5} has transition probability matrix P.
ain {x. " 0) with state spare S-(0 i 2.3.45) I as transition proba- bility matrix 01-α 0 0 1/32/3-3 β/2 0 β/2 0 β/2 β/21/2 0001-γ 0 0 0 0 (a) Determine the equivalence classes of communicating states for any possible choice of the three parameters α, β and γ; (b) In all cases, determine if...
A Markov chain {Xn,n 2 0) with state space S 10, 1, 2,3, 4,5) has transition proba- bility matrix 0 1/32/3-ββ/2 01-α 0 β/2 0 0 0 0 0 0 β/2 β/21/2 0 1. Y (a) Determine the equivalence classes of communicating states for any possible choice of the three parameters α, β and γ; (b) In all cases, determine if the states in each class are recurrent or transient and find their period (or determine that they are aperiodic)
1. A Markov chain (x,, n 2 01 with state space S (0,1,2,3,4,5] has transition proba- bility matrix Γα β/2 01-α 0 0 0 0 1/32/3_ββ/2 β/2 β/2 1/2 0 0 0 0 (a) Determine the equivalence classes of communicating states for any possible choice of the three parameters α, β and γ; (b) In all cases, determine if the states in each class are recurrent or transient and find their period (or determine that they are aperiodic)
4. Consider an irreducible Markov chain with finite state space S = {0, 1, , (a) Starting at state i, what is the probability that it will ever visit state j? (i,j arbi trary (b) Suppose that Xjj iyi for al i. Let ai P(visit N before 0 start at i). Show uations that the r, satisfy, and show that Xi . H2nt: Derive a system of linear eq that xi- solves these equations
Consider the Markov chain on state space {1,2, 3,4, 5, 6}. From 1 it goes to 2 or 3 equally likely. From 2 it goes back to 2. From 3 it goes to 1, 2, or 4 equally likely. From 4 the chain goes to 5 or 6 equally likely. From 5 it goes to 4 or 6 equally likely. From 6 it goes straight to 5. (a) What are the communicating classes? Which are recurrent and which are transient? What...
A Markov chain {Xn, n ≥ 0} with state space S = {0, 1, 2} has transition probability matrix 0.1 0.3 0.6 p = 0.5 0.2 0.3 0.4 0.2 0.4 If P(X0 = 0) = P(X0 = 1) = 0.4 and P(X0 = 2) = 0.2, find the distribution of X2 and evaluate P[X2 < X4].
6. Suppose Xn is a two-state Markov chain with transition probabilities (Xn, Xn+1), n = 0, 1, 2, Write down the state space of the Markov chain Zo, Zi, . . . and determine the transition probability matrix.
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...