Von Mises distribution

von Mises

Probability density function The support is chosen to be [−π,π] with μ = 0
Cumulative distribution function The support is chosen to be [−π,π] with μ = 0
Parameters	${\displaystyle \mu }$ real ${\displaystyle \kappa >0}$
Support	${\displaystyle x\in }$ any interval of length 2π
PDF	${\displaystyle {\frac {e^{\kappa \cos(x-\mu )}}{2\pi I_{0}(\kappa )}}}$
CDF	(not analytic – see text)
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle {\textrm {var}}(x)=1-I_{1}(\kappa )/I_{0}(\kappa )}$ (circular)
Entropy	${\displaystyle -\kappa {\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}+\ln[2\pi I_{0}(\kappa )]}$ (differential)
CF	${\displaystyle {\frac {I_{|t|}(\kappa )}{I_{0}(\kappa )}}e^{it\mu }}$

In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or the Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle ${\displaystyle \theta }$ on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation.^[1] The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the N-dimensional sphere.