Central limit theorem

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In probability theory, the central limit theorem (CLT) states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed (Rice 1995). The central limit theorem also requires the random variables to be identically distributed, unless certain conditions are met. Since real-world quantities are often the balanced sum of many unobserved random events, this theorem provides a partial explanation for the prevalence of the normal probability distribution. The CLT also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.

A simple example of the central limit theorem is given by the problem of rolling a large number of dice, each of which is weighted unfairly in some unknown way. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution, the parameters of which can be determined empirically.

For other generalizations for finite variance which do not require identical distribution, see Lindeberg's condition, Lyapunov's condition, Gnedenko and Kolmogorov states.

In more general probability theory, a central limit theorem is any of a set of weak-convergence theories. They all express the fact that a sum of many independent random variables will tend to be distributed according to one of a small set of "attractor" (i.e. stable) distributions. When the variance of the variables is finite, the "attractor" distribution is the normal distribution. Specifically, the sum of a number of random variables with power law tail distributions decreasing as 1/|x|α + 1 where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution with stability parameter (or index of stability) of α as the number of variables grows.[1] This article is concerned only with the classical (i.e. finite variance) central limit theorem.


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