📂Probability Distribution

Wrapped Normal Distribution

Definition¹

We denote the wrapped distribution of the normal distribution $N(\mu, \sigma^{2})$ by $\operatorname{WN}(\mu, \sigma^{2})$ and call it the wrapped normal distribution. The probability density function of the wrapped normal distribution is given by:

$$ f_{\text{WN}}(\theta; \mu, \sigma) = \sum_{k \in \mathbb{Z}} \dfrac{1}{\sqrt{2\pi\sigma^{2}}} \exp\left( -\dfrac{(\theta - \mu + 2\pi k )^{2}}{2\sigma^{2}} \right), \qquad \theta \in [0, 2\pi) $$

Explanation

Intuitively, the wrapped normal distribution can be viewed as a normal distribution defined on the circle. A related distribution defined by a similar concept is the von Mises distribution. The wrapped normal distribution is obtained by mapping the normal distribution defined on $\mathbb{R}$ to $S^{1}$, whereas the von Mises distribution is originally defined on $S^{1}$.

Properties

(a) $f_{\text{WN}}$ can also be expressed in the following form. $$ f_{\text{WN}}(\theta; \mu, \sigma) = \dfrac{1}{2\pi} \sum_{n \in \mathbb{Z}} e^{-\sigma^{2}n^{2}/2 + \mathrm{i}(\theta - \mu) } $$