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

subject to the following conditions:

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

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

$X≔\mathrm{RandomVariable}\left(\mathrm{Moyal}\left(\mathrm{\mu }\,\mathrm{\sigma }\right)\right)\:$

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

$\frac{\sqrt{2}{\ⅇ}^{-\frac{u-\mathrm{\mu }}{2\mathrm{\sigma }}-\frac{{\ⅇ}^{-\frac{u-\mathrm{\mu }}{\mathrm{\sigma }}}}{2}}}{2\sqrt{\mathrm{\pi }}\mathrm{\sigma }}$

$\mathrm{PDF}\left(X\,0.5\right)$

$\frac{0.3989422802{\ⅇ}^{-\frac{0.5000000000\left(0.5-1.\mathrm{\mu }\right)}{\mathrm{\sigma }}-0.5000000000{\ⅇ}^{-\frac{1.\left(0.5-1.\mathrm{\mu }\right)}{\mathrm{\sigma }}}}}{\mathrm{\sigma }}$

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

$\mathrm{\sigma }\left(\mathrm{\gamma }+\ln \left(2\right)\right)+\mathrm{\mu }$

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

$\frac{{\mathrm{\sigma }}^2{\mathrm{\pi }}^2}{2}$

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.