A drunken man wanders around randomly in a large space. At each step, he moves one unit
Chapter 7, Problem 52(choose chapter or problem)
A drunken man wanders around randomly in a large space. At each step, he moves one unit of distance North, South, East, or West, with equal probabilities. Choose coordinates such that his initial position is (0, 0) and if he is at (x, y) at some time, then one step later he is at (x, y + 1),(x, y 1),(x + 1, y), or (x 1, y). Let (Xn, Yn) and Rn be his position and distance from the origin after n steps, respectively. General hint: Note that Xn is a sum of r.v.s with possible values 1, 0, 1, and likewise for Yn, but be careful throughout the problem about independence. (a) Determine whether or not Xn is independent of Yn. (b) Find Cov(Xn, Yn). (c) Find E(R2 n)
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