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:
Characteristic function
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:
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
Generation of random variates
If X is a random variate drawn from an asymmetric Laplace distribution (ALD), then will be a circular variate drawn from the wrapped ALD, and, will be an angular variate drawn from the wrapped ALD with .
Since the ALD is the distribution of the difference of two variates drawn from the exponential distribution, it follows that if Z1 is drawn from a wrapped exponential distribution with mean m1 and rate λ/κ and Z2 is drawn from a wrapped exponential distribution with mean m2 and rate λκ, then Z1/Z2 will be a circular variate drawn from the wrapped ALD with parameters ( m1 - m2 , λ, κ) and will be an angular variate drawn from that wrapped ALD with .