\begin{definition}[Definition:Generalized Inverse Gaussian Distribution]
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{{Probability distribution|
   name       =Generalized inverse Gaussian|
   type       =density|
   pdf_image  =|
   cdf_image  =|
   parameters =''a'' > 0, ''b'' > 0, ''p'' real|
   support    =''x'' > 0|
   pdf        =$f(x) = \frac{(a/b)^{p/2}}{2 K_p(\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2}$|
   cdf        =|
   mean       =$\frac{\sqrt{b}\ K_{p+1}(\sqrt{a b}) }{ \sqrt{a}\ K_{p}(\sqrt{a b})}$|
   median     =|
   mode       =$\frac{(p-1)+\sqrt{(p-1)^2+ab}}{a}$|
   variance   =$\left(\frac{b}{a}\right)\left[\frac{K_{p+2}(\sqrt{ab})}{K_p(\sqrt{ab})}-\left(\frac{K_{p+1}(\sqrt{ab})}{K_p(\sqrt{ab})}\right)^2\right]$|
   skewness   =|
   kurtosis   =|
   entropy    =|
   mgf        =$\left(\frac{a}{a-2t}\right)^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2t})}{K_p(\sqrt{ab})}$|
   char       =$\left(\frac{a}{a-2it}\right)^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2it})}{K_p(\sqrt{ab})}$|
 }}
-->
The '''generalized inverse Gaussian distribution''' ('''GIG''')  is a three-parameter family of continuous probability distributions with probability density function:
:$\forall x > 0: \map f x = \dfrac {\paren {a / b}^{p/2} } {2 \map {K_p} {\sqrt{a b} } } x^{\paren {p - 1} } e^{-\paren {a x + b / x} / 2}$
where:
:$K_p$ is a modified Bessel function of the second kind
:$a > 0, b > 0, p$ are real.
Category:Definitions/Examples of Probability Distributions
\end{definition}