Introduction to Probability · Problem 12.1.2

Random Walks — Grinstead & Snell

Mathematics — Probability · random-walks-return-probabilities

Problem

(a) Show that for m≥1m \geq 1, f2m=u2m−2−u2m.f_{2m} = u_{2m-2} - u_{2m}. (b) Using part (a), find a closed-form expression for the sum f2+f4+⋯+f2m.f_2 + f_4 + \cdots + f_{2m}. (c) Using part (b), show that ∑m=1∞f2m=1.\sum_{m=1}^{\infty} f_{2m} = 1. (One can also obtain this statement from the fact that F(x)=1−(1−x)1/2F(x) = 1 - (1-x)^{1/2}.) (d) Using parts (a) and (b), show that the probability of no equalization in the first 2m2m outcomes equals the probability of an equalization at time 2m2m.

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