Problem 1 Consider a Markov chain with three states S = {1, 2, 3} with the following one-step

transition probability matrix:

P :=

1/2 1/4 1/4

1/2 0 1/2

1/4 1/4 1/2

(a) Is the Markov chain irreducible and aperiodic? Explain why or why not.

(b) Starting from state 1, what is the long-run proportion of time that the Markov chain will be

in state 2?

P