Given a Poisson distribution with rate of change 
,
the distribution of waiting times between successive changes (with 
) is
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
and the probability distribution function is
 |
(4)
|
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)
|
The raw moments are given by
 |
(6)
|
the first few of which are therefore 1, 
, 
, 
, 
, .... Similarly, the central
moments are
 |  |  |
(7)
|
 |  |  |
(8)
|
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
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(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
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(12)
|
The characteristic function is
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(13)
|
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(14)
|
where 
is the Heaviside step function and ](/images/equations/ExponentialDistribution/Inline47.svg)
is the Fourier transform with parameters 
.
If a generalized exponential probability function is defined by
 |
(15)
|
for 
,
then the characteristic function is
 |
(16)
|
The central moments are
 |
(17)
|
and the raw moments are
 |  |  |
(18)
|
 |  |  |
(19)
|
and the mean, variance, skewness, and kurtosis excess are
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
