We characterise the classes of continuous and discrete phase-type dist
ributions in the following way. They are known to be closed under conv
olutions, mixtures, and the unary 'geometric mixture' operation. We sh
ow that the continuous class is the smallest family of distributions t
hat is closed under these operations and contains all exponential dist
ributions and the point mass at zero. An analogous result holds for th
e discrete class. We also show that discrete phase-type distributions
can be regarded as R+-rational sequences, in the sense of automata the
ory. This allows us to view our characterisation of them as a corollar
y of the Kleene-Schutzenberger theorem on the behavior of finite autom
ata. We prove moreover that any summable R+-rational sequence is propo
rtional to a discrete phase-type distribution.