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Wrapped exponential distribution
Probability distribution From Wikipedia, the free encyclopedia
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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.
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Definition
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The probability density function of the wrapped exponential distribution is[1]
for where 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.
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Characteristic function
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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 is the Lerch transcendent function.
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Circular moments
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In terms of the circular variable the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:
where is some interval of length . 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
and the variance is then 1-R.
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Characterisation
The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range for a fixed value of the expectation .[1]
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See also
References
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