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.

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} .}

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

• BuTools a MATLAB and Mathematica script for fitting matrix-exponential distributions to three specified moments.

• Rational arrival process