Editing Normal distribution (section)

=== Central limit theorem ===
[[File:De moivre-laplace.gif|right|thumb|250px|As the number of discrete events increases, the function begins to resemble a normal distribution.]]
[[File:Dice sum central limit theorem.svg|thumb|250px|Comparison of probability density functions, {{math|''p''(''k'')}} for the sum of {{mvar|n}} fair 6-sided dice to show their convergence to a normal distribution with increasing {{mvar|na}}, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).]]
{{Main|Central limit theorem}}

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where <math display=inline>X_1,\ldots ,X_n</math> are [[independent and identically distributed]] random variables with the same arbitrary distribution, zero mean, and variance <math display=inline>\sigma^2</math> and {{tmath|Z}} is their
mean scaled by <math display=inline>\sqrt{n}</math>
<math display=block>Z = \sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i\right)</math>
Then, as {{tmath|n}} increases, the probability distribution of {{tmath|Z}} will tend to the normal distribution with zero mean and variance {{tmath|\sigma^2}}.

The theorem can be extended to variables <math display=inline>(X_i)</math> that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many [[test statistic]]s, [[score (statistics)|scores]], and [[estimator]]s encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of [[influence function (statistics)|influence functions]]. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
* The [[binomial distribution]] <math display=inline>B(n,p)</math> is [[De Moivre–Laplace theorem|approximately normal]] with mean <math display=inline>np</math> and variance <math display=inline>np(1-p)</math> for large {{tmath|n}} and for {{tmath|p}} not too close to 0 or 1.
* The [[Poisson distribution]] with parameter {{tmath|\lambda}} is approximately normal with mean {{tmath|\lambda}} and variance {{tmath|\lambda}}, for large values of {{tmath|\lambda}}.<ref>{{cite web|url=http://www.stat.ucla.edu/~dinov/courses_students.dir/Applets.dir/NormalApprox2PoissonApplet.html|title=Normal Approximation to Poisson Distribution|website=Stat.ucla.edu|access-date=2017-03-03}}</ref>
* The [[chi-squared distribution]] <math display=inline>\chi^2(k)</math> is approximately normal with mean {{tmath|k}} and variance <math display=inline>2k</math>, for large {{tmath|k}}.
* The [[Student's t-distribution]] <math display=inline>t(\nu)</math> is approximately normal with mean 0 and variance 1 when {{tmath|\nu}} is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the [[Berry–Esseen theorem]], improvements of the approximation are given by the [[Edgeworth expansion]]s.

This theorem can also be used to justify modeling the sum of many uniform noise sources as [[Gaussian noise]]. See [[AWGN]].