Editing Probability theory (section)

===Central limit theorem===
{{Main|Central limit theorem}}
The central limit theorem (CLT) explains the ubiquitous occurrence of the [[normal distribution]] in nature, and this theorem, according to David Williams, "is one of the great results of mathematics."<ref>[[David Williams (mathematician)|David Williams]], "Probability with martingales", Cambridge 1991/2008</ref><!-- Why? -->

The theorem states that the [[average]] of many independent and identically distributed random variables with finite variance tends towards a normal distribution ''irrespective'' of the distribution followed by the original random variables. Formally, let <math>X_1,X_2,\dots\,</math> be independent random variables with [[mean]] <math>\mu</math> and [[variance]] <math>\sigma^2 > 0.\,</math> Then the sequence of random variables
:<math>Z_n=\frac{\sum_{i=1}^n (X_i - \mu)}{\sigma\sqrt{n}}\,</math>
converges in distribution to a [[standard normal]] random variable.

For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the [[Berry–Esseen theorem]]. For example, the distributions with finite  first, second, and third moment from the [[exponential family]]; on the other hand, for some random variables of the [[heavy tail]] and [[fat tail]] variety, it works very slowly or may not work at all: in such cases one may use the [[Stable distribution#A generalized central limit theorem|Generalized Central Limit Theorem]] (GCLT).