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
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	${\displaystyle \frac{1}{1-in/\lambda }}$

Inprobability theory anddirectional statistics, awrapped exponential distribution is awrapped probability distribution that results from the "wrapping" of theexponential distribution around theunit circle.

Theprobability density function of the wrapped exponential distribution is^[1]

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

for where${\displaystyle \lambda >0}$ is the rate parameter of the unwrapped distribution. This is identical to thetruncated distribution obtained by restricting observed valuesX from theexponential distribution with rate parameterλ to the range. Note that this distribution is not periodic.

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

${\displaystyle {\phi }_n(\lambda )=\frac{1}{1-in/\lambda }}$

which yields an alternate expression for the wrapped exponential PDF in terms of the circular variablez =e^i(θ-m) valid for all realθ andm:

where${\displaystyle \mathrm{\Phi }()}$ is theLerch 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:

${\displaystyle ⟨z^n⟩={\int }_{\mathrm{\Gamma }}{e}^{in\theta }{f}_{\text{WE}}(\theta ;\lambda )d\theta =\frac{1}{1-in/\lambda },}$

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

${\displaystyle ⟨z⟩=\frac{1}{1-i/\lambda }.}$

The mean angle is

${\displaystyle ⟨\theta ⟩=\mathrm{A}\mathrm{r}\mathrm{g}⟨z⟩=\arctan (1/\lambda ),}$

and the length of the mean resultant is

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

and the variance is then 1 −R.

The wrapped exponential distribution is themaximum entropy probability distribution for distributions restricted to the range for a fixed value of the expectation${\displaystyle \mathrm{E}(\theta )}$.^[1]

• Wrapped distribution
• Directional statistics

• Continuous distributions
• Directional statistics

• Articles with short description
• Short description matches Wikidata