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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

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

 See Also