Wrapped exponential distribution

Wrapped Exponential
Probability density function The support is chosen to be [0,2π]
Cumulative distribution function The support is chosen to be [0,2π]
Parameters	${\displaystyle \lambda >0}$
Support	${\displaystyle 0\leq \theta <2\pi }$
PDF	${\displaystyle {\frac {\lambda e^{-\lambda \theta }}{1-e^{-2\pi \lambda }}}}$
CDF	${\displaystyle {\frac {1-e^{-\lambda \theta }}{1-e^{-2\pi \lambda }}}}$
Mean	${\displaystyle \arctan(1/\lambda )}$ (circular)
Variance	${\displaystyle 1-{\frac {\lambda }{\sqrt {1+\lambda ^{2}}}}}$ (circular)
Entropy	${\displaystyle 1+\ln \left({\frac {\beta -1}{\lambda }}\right)-{\frac {\beta }{\beta -1}}\ln(\beta )}$ where ${\displaystyle \beta =e^{2\pi \lambda }}$ (differential)
CF	${\displaystyle {\frac {1}{1-in/\lambda }}}$

In probability theory and directional statistics, a wrapped exponential distribution is a wrapped probability distribution that results from the "wrapping" of the exponential distribution around the unit circle.

The probability density function of the wrapped exponential distribution is¹

${\displaystyle f_{\text{WE}}(\theta ;\lambda )=\sum _{k=0}^{\infty }\lambda e^{-\lambda (\theta +2\pi k)}={\frac {\lambda e^{-\lambda \theta }}{1-e^{-2\pi \lambda }}},}$

for ${\displaystyle 0\leq \theta <2\pi }$ where ${\displaystyle \lambda >0}$ is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter λ to the range ${\displaystyle 0\leq X<2\pi }$. Note that this distribution is not periodic.

The characteristic function of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:

${\displaystyle \varphi _{n}(\lambda )={\frac {1}{1-in/\lambda }}}$

which yields an alternate expression for the wrapped exponential PDF in terms of the circular variable z = e^i(θ-m) valid for all real θ and m:

${\displaystyle {\begin{aligned}f_{\text{WE}}(z;\lambda )&={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }{\frac {z^{-n}}{1-in/\lambda }}\\[10pt]&={\begin{cases}{\frac {\lambda }{\pi }}\,{\textrm {Im}}(\Phi (z,1,-i\lambda ))-{\frac {1}{2\pi }}&{\text{if }}z\neq 1\\[12pt]{\frac {\lambda }{1-e^{-2\pi \lambda }}}&{\text{if }}z=1\end{cases}}\end{aligned}}}$

where ${\displaystyle \Phi ()}$ is the Lerch transcendent function.

In terms of the circular variable ${\displaystyle z=e^{i\theta }}$ the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:

${\displaystyle \langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{\text{WE}}(\theta ;\lambda )\,d\theta ={\frac {1}{1-in/\lambda }},}$

where ${\displaystyle \Gamma \,}$ is some interval of length ${\displaystyle 2\pi }$. The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

${\displaystyle \langle z\rangle ={\frac {1}{1-i/\lambda }}.}$

The mean angle is

${\displaystyle \langle \theta \rangle =\mathrm {Arg} \langle z\rangle =\arctan(1/\lambda ),}$

and the length of the mean resultant is

${\displaystyle R=|\langle z\rangle |={\frac {\lambda }{\sqrt {1+\lambda ^{2}}}}.}$

and the variance is then 1 − R.

The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range ${\displaystyle 0\leq \theta <2\pi }$ for a fixed value of the expectation ${\displaystyle \operatorname {E} (\theta )}$.¹