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Kaniadakis Gaussian distribution

Continuous probability distribution From Wikipedia, the free encyclopedia

Kaniadakis Gaussian distribution
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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,[1] geophysics,[2] astrophysics, among many others.

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The κ-Gaussian distribution is a particular case of the κ-Generalized Gamma distribution.[3]

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Definitions

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Probability density function

The general form of the centered Kaniadakis κ-Gaussian probability density function is:[3]

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

Cumulative distribution function

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

where

is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function as .

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Properties

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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 and is given by:

Kurtosis

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:

or

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κ-Error function

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Quick Facts κ-Error function, General information ...

The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:[3]

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 .

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Applications

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

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See also

References

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