Let {Mn: n 2 0} be the symmetric random walk of Exercise 1, and define lo = 0 and n- In = _M;(M;+1 - M;), n21. j=0 1. Show that In = n - 2. Let n be an arbitrary non-negative integer, and let f(i) be an arbitrary function of a variable i. In terms of n and f, define another function g(i) satisfying En If (In+1)] = g( In). Note that although the function g(In) on the right-hand side of this equation may depend on n, the only random variable that may appear in its argument is In. The random variable Mn may not appear. Proving this is proving that the stochastic process { In : n 2 0} is a Markov process.Toss a coin repeatedly. Assume the probability of head on each toss is %, as is the probability od tail. Let X j = 1 if the jth toss results in a head and X 3- = 1 if the jth toss results in a tail. Consider the stochastic process {an n 2 0} dened by M0 = 0 and Mn =2Xj, n2 1. i=1 This is called a symmetric random walk: with each head, it steps up one, and with each tail, it steps down one