A Markov chain {Xn,n=0,1,2,...} with state space S = {0,1,2,3} is applied to model a process. The transition probabilities between states in one year are given by: 0.7 0.3 0 0 0.2 0.8 0 0 0.1 0.3 0 0.6 0.4 0 0.6 0 P : (Pg)4x4 : (a) For this Markov chain, determine Whether it is reducible or irreducible, and its states are periodic or aperiodic. [2] (b) Find all stationary distributions of this Markov chain. [3] (0) Determine Whether 31130 P\" and 130 Pr(X,, = i) exist or not and justify your answers. Find the limits that exist. [5] ((1) At the end of each year, the process incurs a cost of 80 if it is in state 0 at the start of the year, 50 if in state 1, and 20 if in state 2 or 3. Given that the process is in state 0 at the start of the rst year, calculate the expected present value of the total cost of the process over all iture years (including the rst year) at interest rate 5% pa. [5]