|Parameters||α, T, s|
|Support||x ∈ [0, ∞)|
|α ex Ts|
|CDF||1 + αexTT−1s|
In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform. They were first introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms.
The probability density function is (and 0 when x < 0), and the cumulative distribution function is  where 1 is a vector of 1s and
There are no restrictions on the parameters α, T, s other than that they correspond to a probability distribution. There is no straightforward way to ascertain if a particular set of parameters form such a distribution. The dimension of the matrix T is the order of the matrix-exponential representation.
The distribution is a generalisation of the phase-type distribution.
If X has a matrix-exponential distribution then the kth moment is given by
Matrix exponential distributions can be fitted using maximum likelihood estimation.