A symmetric random walk in two dimensions is defined to be a sequence of points {(Xn, Yn): n≥ 0} which evolves in the following way: if (Xn, Yn) = (x, y) then (Xn+1, Yn+1) is one of the four points (x ±1, y), (x, y ± 1), each being picked with equal probability. If (Xo, Yo) = (0,0): (a) show that E(X2 + Y2) = n, (b) find the probability po (2n) that the particle is at the origin after the (2n)th step, and deduce that the probability of ever returning to the origin is 1.

A symmetric random walk in two dimensions is defined to be a sequence of points {(Xn, Yn): n≥ 0} which evolves in the following way: if (Xn, Yn) = (x, y) then (Xn+1, Yn+1) is one of the four points (x ±1, y), (x, y ± 1), each being picked with equal probability. If (Xo, Yo) = (0,0): (a) show that E(X2 + Y2) = n, (b) find the probability po (2n) that the particle is at the origin after the (2n)th step, and deduce that the probability of ever returning to the origin is 1.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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