en>Eugene-elgato at 19:44, 16 September 2013

2013-09-16T19:44:22Z

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{{unreferenced|date=June 2013}}
In [[statistics]], a '''symmetric probability distribution''' is a [[probability distribution]]—an assignment of probabilities to possible occurrences—which is unchanged when its [[probability density function]] or [[probability mass function]] is [[reflection (mathematics)|reflected]] around a vertical line at some value of the [[random variable]] represented by the distribution. This vertical line is the line of [[symmetry (mathematics)|symmetry]] of the distribution. Thus the probability of being any given distance on one side of the value about which symmetry occurs is the same as the probability of being the same distance on the other side of that value.

==Formal definition==

A probability distribution is said to be symmetric [[if and only if]] there exists a value <math>x_0</math> such that

:<math> f(x_0-\delta) = f(x_0+\delta) </math> for all real numbers <math>\delta ,</math>

where ''f'' is the probability density function if the distribution is [[continuous distribution|continuous]] or the probability mass function if the distribution is [[discrete distribution|discrete]].

==Properties==

*The [[median]] and the [[mean]] (if it exists) of a symmetric distribution both occur at the point <math>x_0</math> about which the symmetry occurs.

*If a symmetric distribution is [[Unimodal distribution|unimodal]], the [[Mode (statistics)|mode]] coincides with the [[median]].

*All odd [[central moment]]s of a symmetric distribution equal zero (if they exist), because in the calculation of such moments the negative terms arising from negative deviations from <math>x_0</math> exactly balance the positive terms arising from equal positive deviations from <math>x_0</math>.

*Every measure of [[skewness]] equals zero for a symmetric distribution.

==Probability density function==

Typically a symmetric continuous distribution's probability density function contains the index value <math>x</math> only in the context of a term <math>(x-x_0)^{2k}</math> where <math>k</math> is some positive integer (usually 1). This quadratic or other even-powered term takes on the same value for <math>x=x_0 - \delta</math> as for <math>x=x_0 + \delta</math>, giving symmetry about <math>x_0</math>. Sometimes the density function contains the term <math>|x-x_0|</math>, which also shows symmetry about <math>x_0.</math>

==Partial list of examples==

The following distributions are symmetric for all parametrizations. (Many other distributions are symmetric for a particular parametrization.)

*[[Arcsine distribution]]
*[[Bates distribution]]
*[[Cauchy distribution]]
*[[Champernowne distribution]]
*[[Continuous uniform distribution]]
*[[Degenerate distribution]]
*[[Discrete uniform distribution]]
*[[Elliptical distribution]]s
*[[Gaussian q-distribution]]
*[[Generalized normal distribution]]
*[[Hyperbolic secant distribution]]
*[[Irwin–Hall distribution]]
*[[Laplace distribution]]
*[[Logistic distribution]]
*[[Normal distribution]]
*[[Normal-exponential-gamma distribution]]
*[[Rademacher distribution]]
*[[Raised cosine distribution]]
*[[Student's t distribution]]
*[[Tukey lambda distribution]]
*[[U-quadratic distribution]]
*[[Voigt profile|Voigt distribution]]
*[[von Mises distribution]]
*[[Wigner semicircle distribution]]

{{ProbDistributions}}
{{DEFAULTSORT:Probability Distribution}}
[[Category:Probability distributions|*]]

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en>Eugene-elgato at 19:44, 16 September 2013