A Skip Free Negative Random Walk. Consider an iid sequence {Xn} of random variables with values in {1, 0, 1, 2, . For n 1 and j 0, let pj := P[Xn = j...

A Skip Free Negative Random Walk. Consider an iid sequence {Xn} of random variableswith values in {−1, 0, 1, 2, . . .}. For n ≥ 1 and j ≥ 0, let pj := P[Xn = j − 1], where ∞j=0 pj = 1. For s ∈ [−1, 1], let f(s) :=∞j=0 pjsj. Let S0 = X0 = 1, and for n ≥ 1 letSn := X0 + X1 + · · · + Xn. Let N := inf{n : Sn = 0}. For s ∈ [−1, 1], let P(s) := E[sN].1. Show that P(s) = sf(P(s)).2. Consider the geometric distribution with f(s) = p/(1−qs). Find the smallest solutionP(s) of P(s) = sf(P(s)).