  Let $f$ be the density function associated to a matrix-exponential
distribution of parameters $(\alpha, T,s)$. By exponentially tilting $f$, we
find a probabilistic interpretation which generalises the one associated to
phase-type distributions. More specifically, we show that for any sufficiently
large $\lambda\ge 0$, the function $x\mapsto \left(\int_0^\infty e^{-\lambda
r}f(r) dr\right)^{-1}e^{-\lambda x}f(x)$ can be described in terms of a Markov
jump process whose generator is tied to $T$. Finally, we show how to revert the
exponential tilting in order to assign a probabilistic interpretation to $f$
itself.
