2. A car-insurance company classifies drivers in three categories: bad, neutral and good. The reclassification is done in January of each year and the probabilities for transitions between different categories is given by 1/2 1/2 0 P = 1/5 2/5 2/5 1/5 1/5 3/5 where the first row/column corresponds to the bad category, the second to neutral and the third to good. Suppose that the company started in January 1990 with 1400 drivers in each category (i.e., 4200 drivers in total) and the total number of drivers does not change in time. Let X, represent the state (1 =bad, 2 =neutral and 3 =good) of individual driver in 1990 + n year. (a) Explain why {X,)>, is a MC with state space S = {1, 2,3} having the tran- sition probability matrix P. (b) Explain why the MC in (a) is irreducible, aperiodic and positive recurrent. (c) Find the stationary distribution of the MC in (a). (d) Provide an (accurate) expression for the distribution of drivers in each category in 2090