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Parameters Probability density function Cumulative distribution function $0<\kappa <1$ shape (real) $\beta >0$ rate (real) $x\in \mathbb {R}$ $Z_{\kappa }\exp _{\kappa }(-\beta x^{2})\,\,\,;\,\,\,Z_{\kappa }={\sqrt {\frac {2\beta \kappa }{\pi ))}{\Bigg (}1+{\frac {1}{2))\kappa {\Bigg )}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big ))))$ ${\frac {1}{2))+{\frac {1}{2)){\textrm {erf))_{\kappa }{\big (}{\sqrt {\beta ))x{\big )}\$ $0$ $0$ $0$ $\sigma _{\kappa }^{2}={\frac {1}{\beta )){\frac {2+\kappa }{2-\kappa )){\frac {4\kappa }{4-9\kappa ^{2))}\left[{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big )))}\right]^{2)$ $0$ $3\left[{\frac ((\sqrt {\pi ))Z_{\kappa )){2\beta ^{2/3}\sigma _{\kappa }^{4))}{\frac {(2\kappa )^{-5/2)){1+{\frac {5}{2))\kappa )){\frac {\Gamma \left({\frac {1}{2\kappa ))-{\frac {5}{4))\right)}{\Gamma \left({\frac {1}{2\kappa ))+{\frac {5}{4))\right)))-1\right]$ The Kaniadakis Gaussian distribution (also known as κ-Gaussian distribution) is a probability distribution which arises as a generalization of the Gaussian distribution from the maximization of the Kaniadakis entropy under appropriated constraints. It is one example of a Kaniadakis κ-distribution. The κ-Gaussian distribution has been applied successfully for describing several complex systems in economy, geophysics, astrophysics, among many others.

The κ-Gaussian distribution is a particular case of the κ-Generalized Gamma distribution.

## Definitions

### Probability density function

The general form of the centered Kaniadakis κ-Gaussian probability density function is:

$f_{_{\kappa ))(x)=Z_{\kappa }\exp _{\kappa }(-\beta x^{2})$ where $|\kappa |<1$ is the entropic index associated with the Kaniadakis entropy, $\beta >0$ is the scale parameter, and

$Z_{\kappa }={\sqrt {\frac {2\beta \kappa }{\pi ))}{\Bigg (}1+{\frac {1}{2))\kappa {\Bigg )}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big ))))$ is the normalization constant.

The standard Normal distribution is recovered in the limit $\kappa \rightarrow 0.$ ### Cumulative distribution function

The cumulative distribution function of κ-Gaussian distribution is given by

$F_{\kappa }(x)={\frac {1}{2))+{\frac {1}{2)){\textrm {erf))_{\kappa }{\big (}{\sqrt {\beta ))x{\big ))$ where

${\textrm {erf))_{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi ))}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big )))}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt$ is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function ${\textrm {erf))(x)$ as $\kappa \rightarrow 0$ .

## Properties

### Moments, mean and variance

The centered κ-Gaussian distribution has a moment of odd order equal to zero, including the mean.

The variance is finite for $\kappa <2/3$ and is given by:

$\operatorname {Var} [X]=\sigma _{\kappa }^{2}={\frac {1}{\beta )){\frac {2+\kappa }{2-\kappa )){\frac {4\kappa }{4-9\kappa ^{2))}\left[{\frac {\Gamma \left({\frac {1}{2\kappa ))+{\frac {1}{4))\right)}{\Gamma \left({\frac {1}{2\kappa ))-{\frac {1}{4))\right)))\right]^{2)$ ### Kurtosis

The kurtosis of the centered κ-Gaussian distribution may be computed thought:

$\operatorname {Kurt} [X]=\operatorname {E} \left[{\frac {X^{4)){\sigma _{\kappa }^{4))}\right]$ which can be written as

$\operatorname {Kurt} [X]={\frac {2Z_{\kappa )){\sigma _{\kappa }^{4))}\int _{0}^{\infty }x^{4}\,\exp _{\kappa }\left(-\beta x^{2}\right)dx$ Thus, the kurtosis of the centered κ-Gaussian distribution is given by:

$\operatorname {Kurt} [X]={\frac {3{\sqrt {\pi ))Z_{\kappa )){2\beta ^{2/3}\sigma _{\kappa }^{4))}{\frac {|2\kappa |^{-5/2)){1+{\frac {5}{2))|\kappa |)){\frac {\Gamma \left({\frac {1}{|2\kappa |))-{\frac {5}{4))\right)}{\Gamma \left({\frac {1}{|2\kappa |))+{\frac {5}{4))\right)))$ or

$\operatorname {Kurt} [X]={\frac {3\beta ^{11/6}{\sqrt {2\kappa ))}{2)){\frac {|2\kappa |^{-5/2)){1+{\frac {5}{2))|\kappa |)){\Bigg (}1+{\frac {1}{2))\kappa {\Bigg )}\left({\frac {2-\kappa }{2+\kappa ))\right)^{2}\left({\frac {4-9\kappa ^{2)){4\kappa ))\right)^{2}\left[{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big )))}\right]^{3}{\frac {\Gamma \left({\frac {1}{|2\kappa |))-{\frac {5}{4))\right)}{\Gamma \left({\frac {1}{|2\kappa |))+{\frac {5}{4))\right)))$ ## κ-Error function

κ-Error function Plot of the κ-error function for typical κ-values. The case κ=0 corresponds to the ordinary error function.
General information
General definition$\operatorname {erf} _{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi ))}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big )))}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt$ Fields of applicationProbability, thermodynamics
Domain, Codomain and Image
Domain$\mathbb {C}$ Image$\left(-1,1\right)$ Specific features
Root$0$ Derivative${\frac {d}{dx))\operatorname {erf} _{\kappa }(x)=\left(2+\kappa \right){\sqrt {\frac {2\kappa }{\pi ))}{\frac {\Gamma \left({\frac {1}{2\kappa ))+{\frac {1}{4))\right)}{\Gamma \left({\frac {1}{2\kappa ))-{\frac {1}{4))\right)))\exp _{\kappa }(-x^{2})$ The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:

$\operatorname {erf} _{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi ))}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa ))+{\frac {1}{4)){\Big ))){\Gamma {\Big (}{\frac {1}{2\kappa ))-{\frac {1}{4)){\Big )))}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt$ Although the error function cannot be expressed in terms of elementary functions, numerical approximations are commonly employed.

For a random variable X distributed according to a κ-Gaussian distribution with mean 0 and standard deviation ${\sqrt {\beta ))$ , κ-Error function means the probability that X falls in the interval $[-x,\,x]$ .

## Applications

The κ-Gaussian distribution has been applied in several areas, such as: