Given a Poisson distribution with rate of change , the distribution of waiting times between successive changes (with ) is
(1)
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(2)
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(3)
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and the probability distribution function is
(4)
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It is implemented in the Wolfram Language as ExponentialDistribution[lambda].
The exponential distribution is the only continuous memoryless random distribution. It is a continuous analog of the geometric distribution.
This distribution is properly normalized since
(5)
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The raw moments are given by
(6)
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the first few of which are therefore 1, , , , , .... Similarly, the central moments are
(7)
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(8)
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where is an incomplete gamma function and is a subfactorial, giving the first few as 1, 0, , , , , ... (OEIS A000166).
The mean, variance, skewness, and kurtosis excess are therefore
(9)
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(10)
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(11)
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(12)
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The characteristic function is
(13)
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(14)
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where is the Heaviside step function and is the Fourier transform with parameters .
If a generalized exponential probability function is defined by
(15)
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for , then the characteristic function is
(16)
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The central moments are
(17)
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and the raw moments are
(18)
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(19)
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and the mean, variance, skewness, and kurtosis excess are
(20)
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(21)
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(22)
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(23)
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