Past exam of the options I course of the University of Oxford / 2011 / M / 2 / b / Solution

A stationary distribution is a probability row vector $\pi$ satisfying $\pi P=\pi$. Its coordinates obey ${\pi }_{-}(1-p)={\pi }_{+}q$ and ${\pi }_{-}+{\pi }_{+}=1$. ThusBecause both transitions between distinct states have positive probability, the finite Markov chain is irreducible; its positive self-transition probabilities also make it aperiodic.