Discrete Phase-type Distribution

Characterization

Fix a terminating Markov chain. Denote the upper-left block of its transition matrix and the initial distribution. The distribution of the first time to the absorbing state is denoted or .

Its cumulative distribution function is

$F(k)=1-\boldsymbol{\tau}{T}^{k}\mathbf{1},$

for, and its density function is

$f(k)=\boldsymbol{\tau}{T}^{k-1}\mathbf{T^{0}},$

for . It is assumed the probability of process starting in the absorbing state is zero. The factorial moments of the distribution function are given by,

$E=n!\boldsymbol{\tau}(I-{T})^{-n}{T}^{n-1}\mathbf{1},$

where is the appropriate dimension identity matrix.

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Discrete Phase-type Distribution - Characterization