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Statistics[Distributions]

 Maxwell
 Maxwell distribution

 Calling Sequence Maxwell(alpha) MaxwellDistribution(alpha)

Parameters

 alpha - scale parameter

Description

 • The Maxwell 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{1}{\mathrm{\pi }}}{t}^{2}{ⅇ}^{-\frac{{t}^{2}}{2{\mathrm{\alpha }}^{2}}}}{{\mathrm{\alpha }}^{3}}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0<\mathrm{\alpha }$

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

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Maxwell}\left(\mathrm{α}\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{\sqrt{{2}}{}\sqrt{\frac{{1}}{{\mathrm{π}}}}{}{{u}}^{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}\frac{{{u}}^{{2}}}{{{\mathrm{α}}}^{{2}}}}}{{{\mathrm{α}}}^{{3}}}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.1994711401}{}{{ⅇ}}^{{-}\frac{{0.1250000000}}{{{\mathrm{α}}}^{{2}}}}}{{{\mathrm{α}}}^{{3}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{2}{}\sqrt{{2}}{}{\mathrm{α}}}{\sqrt{{\mathrm{π}}}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{{\mathrm{α}}}^{{2}}{}\left({3}{}{\mathrm{π}}{-}{8}\right)}{{\mathrm{π}}}$ (4)

References

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