## Problem of the Day #223: A Not-So-Random WalkOctober 28, 2011

Posted by Billy in : potd , trackback

Albert is currently located at the number $0$ on the number line. Every turn, Albert has a $\frac{1}{x}$ chance of moving one unit in the positive direction and a $1-\frac{1}{x}$ of standing still, where $x$ is the number Albert is currently on. What is Albert’s expected distance from the origin after $20$ moves?

1. Patrick - October 31, 2011

I believe this is simply a Binomial distribution (assuming independence) where we can count a success as moving one unit in the positive direction and a failure as standing still. Thus p = 1/x and E(X)=n*p=20*1/x.

2. Jonathan - November 2, 2011

I get ~= 6.55 with some numerics. I have an idea for a closed form solution that I’m trying out now. Does somebody have one?