# second moment distribution of random walk

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 $$$$ where $$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.