Let X(t) be a continuous time Markov chain with state space {0, 1, 2,3, 4}. So the amount of time process stayed at each state is exponentially distributed. Let Pj (t) = P(X (t) = j|X(0) =i). Suppose the infinitesimal generator of P(t) is the matrix, 1 O O -5 P(t) - I 1 ON 2 A = lim 0 4 0 h-0 h 2.5 -6 0.5 O ! 0 L (a) Let 7 = min{t 2 0 : X(t) # X(0)} be the time of the first jump. Find E[~|X (0) = 1]. Show all of the work and explain crucial steps. (b) Let T = min{t 2 0 : X(t) # X(0)} the time of the first jump. Find P(X(7) = 2 X(0) =3). Show all of the work and explain crucial steps. (c) Let T = min{t 2 0: X(t) =0 or X(t) = 4) be the first time continuous time Markov chain enters 0 or 4. And define 1; = P(X (T) = 0|X(0) = i) Use first step analysis, set up a system of equations for u1, u2, us. Explain all terms you used in your equations but NO NEED TO SOLVE THE EQUATION. (d) Find the stationary distribution of the process