1. (Weather Forecast Model) Suppose that we want to construct a Markov chain model for the weather of a town located somewhere in the Bluegrass region. You are given the following information. c There are three states: rain, snow, and nice. o If they have nice or rain, they have an even chance of having the same the next day. 0 If they have a snow day, they are just as likely to have nice as rain the next day. 0 They never have snow two days in a row. a If there is change from nice or rain, only half of the time is this a change to a snow day. With the above information construct a Markov chain to model the weather of this town. (a) Find the transition matrix P of this model. (b) If you start with rain. What is the probability of a nice day after ve days? (c) If you start with rain. What is the probability of a nice day after fty days? (d) Determine if P is a regular matrix. (e) Find the stationary distribution for this Markov chain. (f) Interpret the stationary distribution of this chain as long-run probabilities. (g) Find the eigenvalues of P