Two-State Markov Chain Stationary Distribution

The stationary distribution $\pi$ satisfies $\pi P=\pi$ and ${\pi }_1+{\pi }_2=1$.

For a two-state chain, use the balance equation between the two transition rates.

1. 1. Set up the balance equation

   The detailed balance condition for a two-state chain gives ${\pi }_1(0.2)={\pi }_2(0.3)$, so ${\pi }_1/{\pi }_2=0.3/0.2=3/2$.

2. 2. Normalize

   Using ${\pi }_1+{\pi }_2=1$ together with the ratio ${\pi }_1=1.5{\pi }_2$ gives the stationary distribution.

   $${\pi }_2=0.4,{\pi }_1=0.6$$

Answer: (0.6, 0.4)