The probability density function of the wrapped asymmetric Laplace distribution is:[1]
where is the asymmetric Laplace distribution. The angular parameter is restricted to . The scale parameter is which is the scale parameter of the unwrapped distribution and is the asymmetry parameter of the unwrapped distribution.
The cumulative distribution function is therefore:
The characteristic function of the wrapped asymmetric Laplace is just the characteristic function of the asymmetric Laplace function evaluated at integer arguments:
which yields an alternate expression for the wrapped asymmetric Laplace PDF in terms of the circular variable z=ei(θ-m) valid for all real θ and m:
where is the Lerch transcendent function and coth() is the hyperbolic cotangent function.
In terms of the circular variable the circular moments of the wrapped asymmetric Laplace distribution are the characteristic function of the asymmetric Laplace distribution evaluated at integer arguments:
The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:
The mean angle is
and the length of the mean resultant is
The circular variance is then 1 − R