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Statistics[Distributions]

 VonMises
 von Mises distribution

 Calling Sequence VonMises(b, a) VonMisesDistribution(b, a)

Parameters

 b - shape parameter a - distribution mode

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t

 subject to the following conditions:

$0

 • The von Mises variate with location parameter a and scale parameter b tending to 0 from the right, tends to the Uniform variate Uniform(a - Pi, a + Pi).
 • Note that the VonMises command is inert and should be used in combination with the RandomVariable command.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{VonMises}\left(b,a\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{a}{-}{\mathrm{\pi }}\\ \frac{{{ⅇ}}^{{b}{}{\mathrm{cos}}{}\left({a}{-}{u}\right)}}{{2}{}{\mathrm{\pi }}{}{\mathrm{BesselI}}{}\left({0}{,}{b}\right)}& {u}{\le }{a}{+}{\mathrm{\pi }}\\ {0}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,\frac{\mathrm{\pi }}{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}-\frac{\mathrm{\pi }}{2}
 $\frac{{{ⅇ}}^{{b}{}{\mathrm{sin}}{}\left({a}\right)}}{{2}{}{\mathrm{\pi }}{}{\mathrm{BesselI}}{}\left({0}{,}{b}\right)}$ (2)
 > $\mathrm{Mode}\left(X\right)$
 ${a}$ (3)
 > $\mathrm{Mean}\left(X\right)$
 ${a}$ (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.