Consider the Markov chain with states 0, 1,.,n and transition prob- abilities Po = 1 = Pan-1, Pp 1 P,-1, 0 Starting at state 0, say that an excursion ends when the chain either returns to 0 or reaches state n Let X, denote the number of transitions in the jth excursion (that is, the one that begins at the jth return to 0), j1

(d) Find E[X,] (Hint. Relate it to the mean time of a gambler's ruin problem)

(e) Let N denote the first excursion that ends in state n, and find E[N]. on

(f) Find (g) Find ,