Stochastic Matrix - Example: The Cat and Mouse - Phase-type Representation

Phase-type Representation

As the game has an absorbing state 5 the distribution of time to absorption is discrete phase-type distributed. Suppose the system starts in state 2, represented by the vector . To simplify the calculations, state five can be ignored. Let

and remove state five to make a sub-stochastic matrix,

T=\begin{bmatrix} 0 & 0 & 1/2 & 0\\ 0 & 0 & 1 & 0\\ 1/4 & 1/4 & 0 & 1/4\\ 0 & 0 & 1/2 & 0\\ \end{bmatrix}\,, with (I-T)^{-1}\boldsymbol{1} =\begin{bmatrix}2.75 \\ 4.5 \\ 3.5 \\ 2.75\end{bmatrix}\,,

where is the identity matrix, and represents a column matrix of all ones. The expected time of the mouse's survival is given by

Higher order moments are given by

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