Editing Stochastic matrix (section)

===Long-term averages===

No matter what the initial state, the cat will eventually catch the mouse (with probability 1) and a stationary state '''&pi;''' = (0,0,0,0,1) is approached as a limit. To compute the long-term average or expected value of a stochastic variable <math>Y</math>, for each state <math>S_j</math> and time <math>t_k</math> there is a contribution of <math>Y_{j,k}\cdot P(S=S_j,t=t_k)</math>. Survival can be treated as a binary variable with <math>Y=1</math> for a surviving state and <math>Y=0</math> for the terminated state. The states with <math>Y=0</math> do not contribute to the long-term average.