(10) Consider a Markov chain with transition matrix a C d a /0 1/2 0 1/2) b1 P = C 1 d \o 1 Identify the communication classes, their periods, and whether the class is recurrent or transient. Find canonical decomposition.

Let (X0, X1, X2, . . .) be the discrete-time, homogeneous Markov chain on state space S = {1, 2, 3, 4, 5, 6} with X0 = 1 and transition matrix

C)

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

Suppose the transition matrix for a Markov Chain is T = stable population, i.e. an x0₂ such that Tx = x. ساله داد Find a non-zero