📂Probability Distribution

Wrapped Distribution

Introduction¹

Let $X$ be a random variable on the real line $\mathbb{R}$. Suppose that $X$ has periodicity. (Note that it is the random variable, not the probability value, that has periodicity.) A random variable with periodicity $\Theta$ is represented as follows.

$$ \theta = x \pmod{2\pi} \tag{1} $$

Intuitively, this is like wrapping the real space $\mathbb{R}$ around the unit circle. One draws the probability density function on an infinitely long tape and wraps it around a cylinder.

If $\theta$ is sampled from $\Theta$, then from $X$ one of $\theta + 2\pi k$ ($k \in \mathbb{Z}$) is sampled. In other words, if we denote the probability for $\Theta$ by $P_{w}$, the probability that $\Theta$ is sampled between $0$ and $\theta$ equals the sum, over all $k$, of the probabilities that $X$ is sampled between $2\pi k$ and $\theta + 2\pi k$. Expressed mathematically:

$$ P_{w}([0, \theta]) = \sum_{k \in \mathbb{Z}} P([2\pi k, \theta + 2\pi k]) $$

Here $P$ is the probability for $X$. Therefore the cumulative distribution function $F_{w}$ of $\Theta$ is expressed in terms of the cumulative distribution function $F$ of $X$ as follows.

$$ F_{w}(\theta) = \sum_{k \in \mathbb{Z}} F(\theta + 2\pi k) - F(2\pi k) $$

Then the probability density function is given by:

$$ f_{w}(\theta) = \sum_{k \in \mathbb{Z}} f(\theta + 2\pi k) $$

Definition

For a random variable $X$ on the real line $\mathbb{R}$, the distribution followed by the random variable $\Theta$ defined by $\Theta = X (\bmod 2\pi)$ is called the wrapped distribution of $X$. The probability density function $f_{w}$ of the wrapped distribution is expressed in terms of the probability density function $f$ of $X$ as follows.

$$ f_{w}(\theta) = \sum_{k \in \mathbb{Z}} f(\theta + 2\pi k) $$

Explanation

A visualization of the wrapped distribution is shown in the animated GIF below.

Properties

(a) For the mapping $x \mapsto x_{w} = x (\bmod{2\pi})$ the following holds. $$ (x + y)_{w} = x_{w} + y_{w} $$