You can find anything on the web. Earlier today, I noticed that the Sterling approximation and the Gaussian distribution both had a $\sqrt{2 \pi}$ in them, so I started thinking that maybe you could apply Sterling’s approximation to the binomial distribution with $p=1/2$ and large $n$ and the central limit theorem to derive the Gaussian distribution. After making a few calculations it seemed to be working and I thought, “Hey, maybe someone else has done this.” Sure enough, Jake Hoffman did it and posted it here. The internet is pretty cool.

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