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Statistics[Distributions]

 Moyal
 Moyal distribution

 Calling Sequence Moyal(mu, sigma) MoyalDistribution(mu, sigma)

Parameters

 mu - mode parameter sigma - scale parameter

Description

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

$f\left(t\right)=\frac{\sqrt{2}{ⅇ}^{-\frac{t-\mathrm{\mu }}{2\mathrm{\sigma }}-\frac{{ⅇ}^{-\frac{t-\mathrm{\mu }}{\mathrm{\sigma }}}}{2}}}{2\sqrt{\mathrm{\pi }}\mathrm{\sigma }}$

 subject to the following conditions:

$\mathrm{\mu }::\mathrm{real},0<\mathrm{\sigma }$

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

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Moyal}\left(\mathrm{μ},\mathrm{σ}\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}\frac{{1}}{{2}}{}\frac{{u}{-}{\mathrm{μ}}}{{\mathrm{σ}}}{-}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}\frac{{u}{-}{\mathrm{μ}}}{{\mathrm{σ}}}}}}{\sqrt{{\mathrm{π}}}{}{\mathrm{σ}}}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{0.3989422802}{}{{ⅇ}}^{{-}\frac{{0.5000000000}{}\left({0.5}{-}{1.}{}{\mathrm{μ}}\right)}{{\mathrm{σ}}}{-}{0.5000000000}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5}{-}{1.}{}{\mathrm{μ}}\right)}{{\mathrm{σ}}}}}}{{\mathrm{σ}}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{σ}}{}\left({\mathrm{γ}}{+}{\mathrm{ln}}{}\left({2}\right)\right){+}{\mathrm{μ}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{1}}{{2}}{}{{\mathrm{σ}}}^{{2}}{}{{\mathrm{π}}}^{{2}}$ (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.