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 is[1] for

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

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: 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: where

the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments: where