[pzhon]

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But consider the different but related random variable Y, which represents the maximum distance from the origin over the entire walk. That is, the farthest away from 0 that we ever get at any point during the N steps.

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There are two things you might mean by that. One possibility is that you are allowing the distance to be negative, so you want the distribution of the maximum. (This is suggested by your assertion that the distance is normally distributed.) Another possibility is that you are asking what the probability is that the random walk stays between -x and x. Both can be solved by using a reflection method, but before I write up something complicated, which one is your question?