Wrapped normal distribution

In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.

Wrapped Normal

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 ${\displaystyle \sigma >0}$
Support	${\displaystyle \theta \in }$ any interval of length 2π
PDF	${\displaystyle {\frac {1}{2\pi }}\vartheta \left({\frac {\theta -\mu }{2\pi }},{\frac {i\sigma ^{2}}{2\pi }}\right)}$
Mean	${\displaystyle \mu }$ if support is on interval ${\displaystyle \mu \pm \pi }$
Median	${\displaystyle \mu }$ if support is on interval ${\displaystyle \mu \pm \pi }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle 1-e^{-\sigma ^{2}/2}}$ (circular)
Entropy	(see text)
CF	${\displaystyle e^{-\sigma ^{2}n^{2}/2+in\mu }}$