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.
Definition The probability density function of the wrapped exponential distribution is
f WE ( θ ; λ ) = ∑ k = 0 ∞ λ e − λ ( θ + 2 π k ) = λ e − λ θ 1 − e − 2 π λ , {\displaystyle f_{\text{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 ≤ θ < 2 π {\displaystyle 0\leq \theta <2\pi } where λ > 0 {\displaystyle \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 ≤ X < 2 π {\displaystyle 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:
φ n ( λ ) = 1 1 − i n / λ {\displaystyle \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 = ei(θ-m) valid for all real θ and m:
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