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
 |  |  | (9) |
 |  |  | (10) |
 |  |  | (11) |
 |  |  | (12) |
The characteristic function is
 |  |  | (13) |
 |  |  | (14) |
where 
is the Heaviside step function and ](https://mathworld.wolfram.com/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) |
