In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere. In one dimension, a wrapped distribution consists of points on the unit circle. If ${\displaystyle \phi }$ is a random variate in the interval ${\displaystyle (-\infty ,\infty )}$ with probability density function (PDF) ${\displaystyle p(\phi )}$, then ${\displaystyle z=e^{i\phi }}$ is a circular variable distributed according to the wrapped distribution ${\displaystyle p_{wz}(\theta )}$ and ${\displaystyle \theta =\arg(z)}$ is an angular variable in the interval ${\displaystyle (-\pi ,\pi ]}$ distributed according to the wrapped distribution ${\displaystyle p_{w}(\theta )}$.

Any probability density function ${\displaystyle p(\phi )}$ on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the PDF of the wrapped variable

is

which is a periodic sum of period ${\displaystyle 2\pi }$. The preferred interval is generally ${\displaystyle (-\pi <\theta \leq \pi )}$ for which ${\displaystyle \ln(e^{i\theta })=\arg(e^{i\theta })=\theta }$.

In most situations, a process involving circular statistics produces angles (${\displaystyle \phi }$) which lie in the interval ${\displaystyle (-\infty ,\infty )}$, and are described by an "unwrapped" probability density function ${\displaystyle p(\phi )}$. However, a measurement will yield an angle ${\displaystyle \theta }$ which lies in some interval of length ${\displaystyle 2\pi }$ (for example, 0 to ${\displaystyle 2\pi }$). In other words, a measurement cannot tell whether the true angle ${\displaystyle \phi }$ or a wrapped angle ${\displaystyle \theta =\phi +2\pi a}$, where ${\displaystyle a}$ is some unknown integer, has been measured.

If we wish to calculate the expected value of some function of the measured angle it will be:

We can express the integral as a sum of integrals over periods of ${\displaystyle 2\pi }$:

Changing the variable of integration to ${\displaystyle \theta '=\phi -2\pi k}$ and exchanging the order of integration and summation, we have