Parameters α, T, s x ∈ [0, ∞) α ex Ts 1 + αexTT−1s

In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform.[1] They were first introduced by David Cox in 1955 as distributions with rational Laplace–Stieltjes transforms.[2]

The probability density function is ${\displaystyle f(x)=\mathbf {\alpha } e^{x\,T}\mathbf {s} {\text{ for ))x\geq 0}$ (and 0 when x < 0), and the cumulative distribution function is ${\displaystyle F(t)=1-\alpha e^((\textbf {A))t}{\textbf {1))}$[3] where 1 is a vector of 1s and

{\displaystyle {\begin{aligned}\alpha &\in \mathbb {R} ^{1\times n},\\T&\in \mathbb {R} ^{n\times n},\\s&\in \mathbb {R} ^{n\times 1}.\end{aligned))}

There are no restrictions on the parameters α, T, s other than that they correspond to a probability distribution.[4] There is no straightforward way to ascertain if a particular set of parameters form such a distribution.[2] The dimension of the matrix T is the order of the matrix-exponential representation.[1]

The distribution is a generalisation of the phase-type distribution.

## Moments

If X has a matrix-exponential distribution then the kth moment is given by[2]

${\displaystyle \operatorname {E} (X^{k})=(-1)^{k+1}k!\mathbf {\alpha } T^{-(k+1)}\mathbf {s} .}$

## Fitting

Matrix exponential distributions can be fitted using maximum likelihood estimation.[5]