The exponential distribution with rate 
is the distribution of waiting times between successive
events in a homogeneous Poisson process. For 
,
its distribution function is
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
and, for 
,
its probability density function is
 |
(4)
|
Both 
and 
are zero for 
.
It is implemented in the Wolfram Language as ExponentialDistribution[lambda].
The exponential distribution is the only nondegenerate continuous distribution with support 
having the memoryless
property. It is a continuous analog of the geometric
distribution.
This distribution is properly normalized since
 |
(5)
|
For a nonnegative integer 
, 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
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
The characteristic function is
 |  |  |
(13)
|
 |  |  |
(14)
|
where 
is the Heaviside step function and ](/images/equations/ExponentialDistribution/Inline53.svg)
is the Fourier transform with parameters 
.
For a real number 
and 
, a generalized exponential probability
density function is defined by
 |
(15)
|
for 
and is zero otherwise. Its characteristic
function is
 |
(16)
|
The raw moments are
 |
(17)
|
and the central moments are
 |  |  |
(18)
|
 |  |  |
(19)
|
and the mean, variance, skewness, and kurtosis excess are
 |  |  |
(20)
|
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
