Phase-type distribution

A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occurs may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state. Each of the states of the Markov process represents one of the phases.

Phase-type

Parameters	${\displaystyle S,\;m\times m}$ subgenerator matrix ${\displaystyle {\boldsymbol {\alpha }}}$, probability row vector
Support	${\displaystyle x\in [0;\infty )\!}$
PDF	${\displaystyle {\boldsymbol {\alpha }}e^{xS}{\boldsymbol {S}}^{0}}$ See article for details
CDF	${\displaystyle 1-{\boldsymbol {\alpha }}e^{xS}{\boldsymbol {1}}}$
Mean	${\displaystyle -{\boldsymbol {\alpha }}{S}^{-1}\mathbf {1} }$
Median	no simple closed form
Mode	no simple closed form
Variance	${\displaystyle 2{\boldsymbol {\alpha }}{S}^{-2}\mathbf {1} -({\boldsymbol {\alpha }}{S}^{-1}\mathbf {1} )^{2}}$
MGF	${\displaystyle -{\boldsymbol {\alpha }}(tI+S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{0}}$
CF	${\displaystyle -{\boldsymbol {\alpha }}(itI+S)^{-1}{\boldsymbol {S}}^{0}+\alpha _{0}}$

It has a discrete-time equivalent – the discrete phase-type distribution.

The set of phase-type distributions is dense in the field of all positive-valued distributions, that is, it can be used to approximate any positive-valued distribution.