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 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)
