Central Limit Theorem

Averages of large collections of random numbers tend to follow a Gaussian or normal distribution in which the probability of getting value x is

Exp[-(x - μ)^{2}/(2 σ^{2})] / (Sqrt[2 π] σ)

The mean μ and standard deviation σ are determined by properties of the random numbers, but the form of the distribution is always the same. The only conditions are that the random numbers should be statistically independent, and that their distribution should have bounded variance, so that, for example, the probability for very large numbers is rapidly damped. (The limit of an infinite collection of numbers gives σ0 in accordance with the law of large numbers.) The pictures at the top of the next page show how averages of successively larger collections of uniformly distributed numbers converge to a Gaussian distribution.

The Central Limit Theorem leads to a self-similarity property for the Gaussian distribution: if one takes n numbers that follow Gaussian distributions, then their average should also follow a Gaussian distribution, though with a standard deviation that is 1/√n times smaller.