Properties of a Simple Random Walk with Boundaries
This Demonstration shows several basic properties of random walks on a one-dimensional lattice of points . At each step, the probability of moving to the right or left is and , respectively; the walk ends when it reaches or . As a function of the starting position , we show the probability of the walk ending at the right boundary, and the average number of steps taken during the walk (regardless of which boundary it eventually hits).
The formulas for these quantities can be found by solving second-order difference equations.
G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed., Oxford: Oxford University Press, 2001.