3. Consider a time-homogeneous DTMC with finite state space {0, 1, ..., S} and corresponding transition matrix P. Furthermore, suppose that each time the Markov chain visits state j, you receive a 'reward' of f(7) units for some function f : {0, ..., S} -> RR. Defining to be the expected total reward given that the Markov chain starts in state i, answer the following questions: (i) Show that , (i) is given by the formula TE Tn(i) = EEPif(). k=0 j=0 (ii) Write your formula from Part (a) in vector format