 InverseGaussian - Maple Help

Statistics[Distributions]

 InverseGaussian
 inverse Gaussian (Wald) distribution Calling Sequence InverseGaussian(mu, lambda) InverseGaussianDistribution(mu, lambda) Parameters

 mu - distribution mean lambda - scale parameter Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{\sqrt{2}\sqrt{\frac{\mathrm{\lambda }}{\mathrm{\pi }{t}^{3}}}{ⅇ}^{-\frac{\mathrm{\lambda }{\left(t-\mathrm{\mu }\right)}^{2}}{2{\mathrm{\mu }}^{2}t}}}{2}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0<\mathrm{\mu },0<\mathrm{\lambda }$

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

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{InverseGaussian}\left(\mathrm{\mu },\mathrm{\lambda }\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{\sqrt{{2}}{}\sqrt{\frac{{\mathrm{\lambda }}}{{\mathrm{\pi }}{}{{u}}^{{3}}}}{}{{ⅇ}}^{{-}\frac{{\mathrm{\lambda }}{}{\left({u}{-}{\mathrm{\mu }}\right)}^{{2}}}{{2}{}{{\mathrm{\mu }}}^{{2}}{}{u}}}}{{2}}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 ${1.128379166}{}\sqrt{{\mathrm{\lambda }}}{}{{ⅇ}}^{{-}\frac{{1.000000000}{}{\mathrm{\lambda }}{}{\left({0.5}{-}{1.}{}{\mathrm{\mu }}\right)}^{{2}}}{{{\mathrm{\mu }}}^{{2}}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{\mu }}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{{\mathrm{\mu }}}^{{3}}}{{\mathrm{\lambda }}}$ (4) References

 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.