Matrix-exponential distribution

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

${\displaystyle \begin{array}{cc}\alpha & \in {\mathbb{R}}^{1\times n},\\ T& \in {\mathbb{R}}^{n\times n},\\ s& \in {\mathbb{R}}^{n\times 1}.\end{array}}$

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.

Moments

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

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