Suppose that X = (X0, X1, X2, . . . ) = {Xn}n=0 is a Markov Chain on a ?nite state space S. For any y ? S, and sequence (x0, x1, . . . ) with xi ? S, let

Question 1.2 (10 pts. Passage/Return Times). Suppose that X = (Xo, X1, X2, -..) = {X,}, is a Markov chain on a finite state space S. For any y e S, and sequence (co, $1, .. . ) with , E S, let Ry (zo, $1, ...) := min {n 2 1:1, = y} so that Ry (X) = min {n 2 1 : X, =y}- As usual we use the convention that the minimum of the empty set is co.] We refer to R (X) as the first passage time. Write a clear explanation why (1.3) Ry (r, X) = Ry (z, Xo. Xi, X2. ... )=101xoxyRy (X) for all I, YES