Inverse Gaussian distribution

Inverse Gaussian

Probability density function
Parameters	${\displaystyle \lambda >0}$ ${\displaystyle \mu >0}$
Support	${\displaystyle x\in (0,\infty )}$
PDF	${\displaystyle \left[{\frac {\lambda }{2\pi x^{3}}}\right]^{1/2}\exp \left\{{\frac {-\lambda (x-\mu )^{2}}{2\mu ^{2}x}}\right\}}$
CDF	${\displaystyle \Phi \left({\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}-1\right)\right)}$ ${\displaystyle +\exp \left({\frac {2\lambda }{\mu }}\right)\Phi \left(-{\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}+1\right)\right)}$ where ${\displaystyle \Phi \left(\right)}$ is the standard normal (standard Gaussian) distribution c.d.f.
Mean	${\displaystyle \scriptstyle \mathbf {E} [X]=\mu }$ ${\displaystyle \scriptstyle \mathbf {E} [{\frac {1}{X}}]={\frac {1}{\mu }}+{\frac {1}{\lambda }}}$
Mode	${\displaystyle \mu \left[\left(1+{\frac {9\mu ^{2}}{4\lambda ^{2}}}\right)^{\frac {1}{2}}-{\frac {3\mu }{2\lambda }}\right]}$
Variance	${\displaystyle \scriptstyle \mathbf {Var} [X]={\frac {\mu ^{3}}{\lambda }}}$ ${\displaystyle \scriptstyle \mathbf {Var} [{\frac {1}{X}}]={\frac {1}{\mu \lambda }}+{\frac {2}{\lambda ^{2}}}}$
Skewness	${\displaystyle 3\left({\frac {\mu }{\lambda }}\right)^{1/2}}$
Ex. kurtosis	${\displaystyle {\frac {15\mu }{\lambda }}}$
MGF	${\displaystyle \exp \left[{{\frac {\lambda }{\mu }}\left(1-{\sqrt {1-{\frac {2\mu ^{2}t}{\lambda }}}}\right)}\right]}$
CF	${\displaystyle \exp \left[{{\frac {\lambda }{\mu }}\left(1-{\sqrt {1-{\frac {2\mu ^{2}\mathrm {i} t}{\lambda }}}}\right)}\right]}$

${\displaystyle \Phi \left({\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}-1\right)\right)}$ ${\displaystyle +\exp \left({\frac {2\lambda }{\mu }}\right)\Phi \left(-{\sqrt {\frac {\lambda }{x}}}\left({\frac {x}{\mu }}+1\right)\right)}$

In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞).