Given the Markov Chain with state probability matrix $$P = \begin{bmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{bmatrix}$$ Prove that this has the stationary distribution $$\pi = [0.6,0.4]$$
When attempting to solve for $\pi P = \pi$ I obtain the following overdetermined system: $$0.8 \pi_1 + 0.3 \pi_2 = \pi_1 \\ \\ \\\ \\ 0.2 \pi_1 + 0.7 \pi_2 = \pi_2$$ and this has infinitely many solutions so I don't know how to proceed.
Best Answer
Solving the system, we have $\pi _1=1.5\pi _2$. Since the two probabilities must sum to 1, $$\pi _1+\pi _2=1\longrightarrow 1.5\pi _2+\pi _2=1\longrightarrow \pi _2=0.4,\:\pi _1=0.6$$