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,
- with
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
Read more about this topic: Stochastic Matrix, Example: The Cat and Mouse