Let {Xn; n 1} be a Markov chain with state space {1, 2, 3} and one-step transition

probability matrix

P

(1) =

0 1 0

1/2 0 1/2

0 1 0

,

where the initial distribution is given by P(X1 = 1) = 1/10, P(X1 = 2) = 6/10 and

P(X1 = 3) = 3/10. Then, answer the following questions.

(a) Is it an irreducible Markov chain?

(b) Is there any ergodic state(s)?

(c) Does there exist stationary distribution(s)? If yes, determine it.

(d) Find P(X2 = 1).