Two steps and the stationary law

Topic: Discrete-time Markov chains · more questions on this topic
A deterministic three-cycle ›

For

P=\begin{pmatrix} 0.9 & 0.1 \\ 0.4 & 0.6 \end{pmatrix},

compute (P^2)_{12} and the stationary distribution.

Worked answer (click to show)

The (1,2) entry of P^2 sums the routes through state 1 and state 2:

(P^2)_{12}=0.9\cdot 0.1+0.1\cdot 0.6=0.15.

Global balance of the single cut is 0.1\pi_1=0.4\pi_2, so \pi_1=4\pi_2. With \pi_1+\pi_2=1, the stationary distribution is (0.8,0.2).