# Kaniadakis Gaussian distribution

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
    $Z_{\kappa }={\sqrt {\frac {2\beta \kappa }{\pi }}}{\Bigg1+{\frac {1}{2}}\kappa {\Bigg )}{\frac {\Gamma {\Big{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}$
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
   ${\textrm {erf}}_{\kappa }(x)={\Big2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma {\Big{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt$
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
   $\operatorname {Kurt} [X]={\frac {3\beta ^{11/6}{\sqrt {2\kappa }}}{2}}{\frac {|2\kappa |^{-5/2}}{1+{\frac {5}{2}}|\kappa |}}{\Bigg1+{\frac {1}{2}}\kappa {\Bigg )}\left({\frac {2-\kappa }{2+\kappa }}\right)^{2}\left({\frac {4-9\kappa ^{2}}{4\kappa }}\right)^{2}\left[{\frac {\Gamma {\Big{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}{\Gamma {\Big{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}}\right]^{3}{\frac {\Gamma \left({\frac {1}{|2\kappa |}}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{|2\kappa |}}+{\frac {5}{4}}\right)}}$

## κ-Error function

The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:
    $\operatorname {erf} _{\kappa }(x)={\Big2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma {\Big{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt$
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:
* In economy, the κ-Gaussian distribution has been applied in the analysis of financial models, accurately representing the dynamics of the processes of extreme changes in stock prices.
* In inverse problems, Error laws in extreme statistics are robustly represented by κ-Gaussian distributions.
* In astrophysics, stellar-residual-radial-velocity data have a Gaussian-type statistical distribution, in which the K index presents a strong relationship with the stellar-cluster ages.
* In nuclear physics, the study of Doppler broadening function in nuclear reactors is well described by a κ-Gaussian distribution for analyzing the neutron-nuclei interaction.
* In cosmology, for interpreting the dynamical evolution of the Friedmann–Robertson–Walker Universe.
* In plasmas physics, for analyzing the electron distribution in electron-acoustic double-layers and the dispersion of Langmuir waves.