Thanks in advance for your help.

1. Consider a discrete Markov chain with state space {1, 2, 3, 4, 5). The transition matrix is /0.1 0.4 0 0.4 0.1' 0 1 0 P= 0 0.5 0.3 0 0.2 0 0 1 0 10.1 0.4 0.2 0.2 0.1 (a) (2 points) If the chain starts with state 3, what is the probability that it arrives at state 5 after one transition (i.e., one "jump")? If the chain starts with state 4, what is the probability that it arrives at state 5 after 1,000, 000 transitions? (b) (2 points) Is this chain reducible or irreducible? Why? (c) (2 points) Is there a limiting probability? Why? (d) (2 points) How many transient states and recurrent states does this chain have? What are they? (e) (2 points) How many invariant distributions does this chain have? What are they