Gaussian integral justification

Question

So I read on the very nice proof of the Gaussian integral being equal to the root of pi and its application for the normal distribution (in fact the normal distribution is described by the Gaussian function). However, maybe the thoughts and proofs behind are really complex, but what decided that specifically the Gaussian function is able to describe the distribution of any Stochastic value repeated an infinite times? It can be seen in many places like physics, but what is the mathematical proof that this function described the probability distribution and not other functions with similar traits (like symmetry, converge to zero in infinity, etc.)?

Answer 1

This is an extremely important result known as the central limit theorem. The proof isn't terribly difficult either (at least considering how fundamental the theorem is).