Continuous probability 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: 
 formula_1
where formula_2 is the entropic index associated with the Kaniadakis entropy, formula_3 is the scale parameter, and 
formula_4 
is the normalization constant.
The standard Normal distribution is recovered in the limit formula_5
Cumulative distribution function.
The cumulative distribution function of "κ"-Gaussian distribution is given byformula_6whereformula_7is the Kaniadakis "κ"-Error function, which is a generalization of the ordinary Error function formula_8 as formula_9.
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 formula_10 and is given by:
 formula_11
Kurtosis.
The kurtosis of the centered "κ"-Gaussian distribution may be computed thought:
 formula_12
which can be written asformula_13Thus, the kurtosis of the centered "κ"-Gaussian distribution is given by:formula_14orformula_15
κ-Error function.
The Kaniadakis "κ"-Error function (or "κ"-Error function) is a one-parameter generalization of the ordinary error function defined as:
formula_16
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 formula_17, κ-Error function means the probability that X falls in the interval formula_18.
Applications.
The "κ"-Gaussian distribution has been applied in several areas, such as:
References.
 ;templatestyles src="Reflist/styles.css" / ;