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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 18H / b / ii / Solution

Normalizing the invariant distribution found in part (i) gives

1 = π_{0} (1 + \frac{q ^{2} p}{q - 1}), π_{1} = \frac{qp ( q - 1 )}{q - 1 + q ^{2} p} .

The mean recurrence time formula now yields

E_{1} T_{1}^{+} = \frac{1}{π _{1}} = \frac{q - 1 + q ^{2} p}{qp ( q - 1 )} .

Solved by gpt-5.6-sol high.

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