As I was arguing today with someone, the average value of a random walk is not 0. And he wanted me to prove it. Well here is the true value of the average: http://en.wikipedia.org/wiki/Random_walk. The gambler’s ruin problem (in the article) is a good example of random walk phenomena in real life (except for brownian motion). Here is a proof that the distance from 0 actually increases as time goes by: http://www.krellinst.org/UCES/archive/modules/monte/node4.html. If you go up the previous website you find this excellent website on Monte-Carlo methods:
http://www.krellinst.org/UCES/archive/modules/monte/node0.html
