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).

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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.
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