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What is the equation for the probability of a random walk with drift being equal to a specific value after n steps, given a specific standard deviation?

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I assume that you expect motion to be gaussian at each step with 0 mean and $\sigma$ standard deviation and each step is i.i.d. (leave comment if this assumption is incorrect).

In such case the probability distribution is simply the convolution of $n$ gaussians with these moments and can be evaluated for any given point using the normal probability density function.

e.g.: After 3 steps $\phi(x) = \frac{1}{\sqrt{2\pi}\hat\sigma} \exp(-\frac{x^{2}}{2\hat\sigma^2})$, where $\hat\sigma = \sigma\sqrt{3}$.

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