50Stationary distribution, 2 states

A two-state Markov chain has transition matrix $\displaystyle \begin{bmatrix}0.9 & 0.1\\0.5 & 0.5\end{bmatrix}$. Find its stationary distribution. Give the answer to 4 decimal places.

Solve π = πP with π₁+π₂=1: this gives π₁ = 5π₂, and substituting into the normalization gives π₂ = 1/6 ≈ 0.1667, π₁ = 5/6 ≈ 0.8333. Check: 0.1(0.8333)+0.5(0.1667) = 0.1667 ✓.