Editing Stochastic matrix (section)

===Phase-type representation===
[[Image:Mousesurvival.jpg|thumb|right|upright=1.6|The survival function of the mouse. The mouse will survive at least the first time step.]]

As State 5 is an absorbing state, the distribution of time to absorption is [[discrete phase-type distribution|discrete phase-type distributed]]. Suppose the system starts in state 2, represented by the vector <math>[0,1,0,0,0]</math>. The states where the mouse has perished don't contribute to the survival average so state five can be ignored. The initial state and transition matrix can be reduced to,

<math display="block">\boldsymbol{\tau}=[0,1,0,0], \qquad T=\begin{bmatrix}
    0          & 0           & \frac{1}{2}  &   0\\
    0          & 0           & 1            &   0\\
  \frac{1}{4}  & \frac{1}{4} &           0  & \frac{1}{4}\\
    0          & 0           & \frac{1}{2}  &   0
\end{bmatrix},</math> 

and 

<math display="block">(I-T)^{-1}\boldsymbol{1} =\begin{bmatrix}2.75 \\ 4.5 \\ 3.5 \\ 2.75\end{bmatrix},</math>

where <math>I</math> is the [[identity matrix]], and <math>\mathbf{1}</math> represents a column matrix of all ones that acts as a sum over states.

Since each state is occupied for one step of time the expected time of the mouse's survival is just the [[Matrix polynomial#Matrix geometrical series|sum]] of the probability of occupation over all surviving states and steps in time,

<math display="block">E[K]=\boldsymbol{\tau} \left (I+T+T^2+\cdots \right )\boldsymbol{1}=\boldsymbol{\tau}(I-T)^{-1}\boldsymbol{1}=4.5.</math>

Higher order moments are given by

<math display="block">E[K(K-1)\dots(K-n+1)]=n!\boldsymbol{\tau}(I-{T})^{-n}{T}^{n-1}\mathbf{1}\,.</math>