 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2011/m/2/b/solution

π

satisfying

π P = π

. Its coordinates obey

π_{-} (1 - p) = π_{+} q

and

π_{-} + π_{+} = 1

. Thus

π_{-} = \frac{q}{1 - p + q}, π_{+} = \frac{1 - p}{1 - p + q} .

Because both transitions between distinct states have positive probability, the finite Markov chain is irreducible; its positive self-transition probabilities also make it aperiodic.

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