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

subject to the following conditions:

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.

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

$X≔\mathrm{RandomVariable}\left(\mathrm{VonMises}\left(b\,a\right)\right)\:$

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

$\left\{\begin{array}{cc}0& u<a-\mathrm{\pi }\\ \frac{{\&ExponentialE;}^{b\cos \left(a-u\right)}}{2\mathrm{\pi }\mathrm{BesselI}\left(0\&comma;b\right)}& u\le a+\mathrm{\pi }\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;\frac{\mathrm{\pi }}{2}\right)\mathbf{assuming}-\frac{\mathrm{\pi }}{2}<a,a<\frac{3\mathrm{\pi }}{2}$

$\frac{{\&ExponentialE;}^{b\sin \left(a\right)}}{2\mathrm{\pi }\mathrm{BesselI}\left(0\&comma;b\right)}$

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

$a$

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

$a$

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.