Generalized inverse Gaussian distribution

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Generalized inverse Gaussian
Probability function
Cumulative distribution function
Parameters	a>0,b>0,p
Support	x>0
Template:Probability distribution/link	$f(x) = \frac{(a/b)^{p/2}}{2 K_p(\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2}$
Cumulative distribution function (cdf)
Mean	$\frac{\sqrt{b}\ K_{-1-p}(\sqrt{a b}) }{ \sqrt{a}\ K_{p}(\sqrt{a b})}$
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function

$f(x) = \frac{(a/b)^{p/2}}{2 K_p(\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2}, \, x > 0,$

where $K p$ is a modified Bessel function of the third kind and $a > 0$ , $b > 0$ . It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Etienne Halphen^[1] It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution, and Herbert Sichel. It is also known as the Sichel Distribution.

A further extension is the "log generalised inverse Gaussian distribution" which, because of its complexity, requires computers to be useful in practice.

[edit] Notes

^ V. Seshadri (1997): Halphen's laws. In S. Kotz, C. B. Read and D. L. Banks (eds.): Encyclopedia of Statistical Sciences, Update Volume 1, pp. 302 - 306. Wiley, New York.

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