The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of an absorbing Markov chain with one absorbing state. Each of the states of the Markov chain represents one of the phases.
Contents
It has continuous time equivalent in the phase-type distribution.
Definition
A terminating Markov chain is a Markov chain where all states are transient, except one which is absorbing. Reordering the states, the transition probability matrix of a terminating Markov chain with
where
Definition. A distribution on
Characterization
Fix a terminating Markov chain. Denote
Its cumulative distribution function is
for
for
where
Special cases
Just as the continuous time distribution is a generalisation of the exponential distribution, the discrete time distribution is a generalisation of the geometric distribution, for example: