Probability density function
Cumulative distribution function
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.
The general form of the centered Kaniadakis κ-Gaussian probability density function is:
where is the entropic index associated with the Kaniadakis entropy, is the scale parameter, and
is the normalization constant.
The standard Normal distribution is recovered in the limit
The cumulative distribution function of κ-Gaussian distribution is given by
is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function as .
The centered κ-Gaussian distribution has a moment of odd order equal to zero, including the mean.
The variance is finite for and is given by:
The kurtosis of the centered κ-Gaussian distribution may be computed thought:
which can be written as
Thus, the kurtosis of the centered κ-Gaussian distribution is given by:
|Fields of application
|Domain, Codomain and Image
The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:
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 , κ-Error function means the probability that X falls in the interval .
The κ-Gaussian distribution has been applied in several areas, such as: