Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ von Mises distribution ▷ Definition

Definition: Let $X$ be a circular random variable. Then, $X$ is said to follow a von Mises distribution with circular mean $\mu$ and reciprocal dispersion $\kappa$

\[\label{eq:vm} X \sim \mathrm{vM}(\mu, \kappa) \; ,\]

if and only if its probability density function is given by

\[\label{eq:vm-pdf} \mathrm{vM}(x; \mu, \kappa) = \frac{1}{2 \pi I_0(\kappa)} \cdot \exp \left[ \kappa \cos(x-\mu) \right]\]

where $\mu \in \mathbb{R}$, $\kappa > 0$ and $I_0(\kappa)$ is the zeroth-order modified Bessel function of the first kind:

\[\label{eq:vm-bessel} I_0(\kappa) = \frac{1}{2\pi} \int_0^{2\pi} \exp \left[ \kappa \cos(x) \right] \, \mathrm{d}x \; .\]
 
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Metadata: ID: D231 | shortcut: vm | author: JoramSoch | date: 2026-04-21, 15:01.