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The inverse Gaussian distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

Note that the InverseGaussian command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{InverseGaussian}\left(\mathrm{\mu }\,\mathrm{\lambda }\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{\sqrt{2}\sqrt{\frac{\mathrm{\lambda }}{\mathrm{\pi }{u}^3}}{\&ExponentialE;}^{-\frac{\mathrm{\lambda }{\left(u-\mathrm{\mu }\right)}^2}{2{\mathrm{\mu }}^{2}u}}}{2}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$1.128379166\sqrt{\mathrm{\lambda }}{\&ExponentialE;}^{-\frac{1.000000000\mathrm{\lambda }{\left(0.5-1.\mathrm{\mu }\right)}^2}{{\mathrm{\mu }}^2}}$

$\mathrm{Mean}\left(X\right)$

$\mathrm{\mu }$

$\mathrm{Variance}\left(X\right)$

$\frac{{\mathrm{\mu }}^3}{\mathrm{\lambda }}$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.