Wrapped exponential distribution

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.

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	$\lambda >0$
Support	$0\leq \theta <2\pi$
PDF	${\frac {\lambda e^{-\lambda \theta }}{1-e^{-2\pi \lambda }}}$
CDF	${\frac {1-e^{-\lambda \theta }}{1-e^{-2\pi \lambda }}}$
Mean	$\arctan(1/\lambda )$ (circular)
Variance	$1-{\frac {\lambda }{\sqrt {1+\lambda ^{2}}}}$ (circular)
Entropy	$1+\ln \left({\frac {\beta -1}{\lambda }}\right)-{\frac {\beta }{\beta -1}}\ln(\beta )$ where $\beta =e^{2\pi \lambda }$ (differential)
CF	${\frac {1}{1-in/\lambda }}$

Definition

The probability density function of the wrapped exponential distribution is

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

for $0\leq \theta <2\pi$ where $\lambda >0$ 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 $0\leq X<2\pi$. Note that this distribution is not periodic.

Characteristic function

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

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

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:

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

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

Circular moments

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

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

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

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

The mean angle is

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

and the length of the mean resultant is

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

and the variance is then 1-R.

Characterisation

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

See also

• Wrapped distribution
• Directional statistics