The Chapman-Kohnogorov equations are given by P;j,+m = 2 13,233; for all n, m a 0, all a, j s=c Explain what these equations mean in general. Use the matrices P and P2 to illustrate the Chapman-Kolmogorov equations for n = l, m = 2, i = j = 1. That is, write down the various ways to go from 1 to 1 in three steps, organised in the manner described in the Chapman-Kolmogrov equations and relate the probabilities in matrices P and 13"2 to these paths. Use the short-hand notation 1 > 0 } O ) 1 to describe, for example, that the chain starts in state 1, moves to state 0, remains in state 0, and then moves to state 1. What is the probability that, starting from state 0, the process never enters state 1? If the initial distribution is (4 / 5, D, 1 / 5), what is the expected value of X 3? Derive the stationary distribution for this Markov chain