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Considering P(X,N) be the probability of finding a 1 dimensional random walk after N steps at position X. What is the probability distribution of $$<X_N^2>$$ where $X_N$ is the position of the Random Walk after N steps and $<>$ is the mean value with respect to the conventional distribution which is binomial in Random Walk case.

now i tried a bit and i would like to prove that the distribution is something like a gaussian (normal) but I dont know how to do that.

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