# 1D Random Walk, with different step sizes in each direction.

A walker starts at a defined position greater than $0$, say $A$, and then makes a "decision" to walk either "$b$ steps to the right" or walk "$c$ steps to the left." He will choose the first option with probability $p$, and the second option with probability $1-p$. If the walker gets to position $0$ he stops.

I wish to calculate:

• the expectation value of the walker's position after a total of n decisions.
• what happens as n approaches infinite?

Is there an existing formula/theory that can be used to get an analytic solution to this problem?

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$A$ must be a multiple of $\gcd(b,c)$ for the probability to be non-zero. – Sasha Sep 8 '12 at 12:48
Please see answers to this question on biased random walk and generalization. – Sasha Sep 8 '12 at 12:51
Thanks for that link Sasha, it does provide some insight on how to solve this problem (i.e. starting at 1, with step sizes of 2 and -1). Unfortunately I believe the method presented becomes more complicated when the starting position exceeds the largest step size, but it was helpful as a starting point. – Mew Sep 12 '12 at 1:53