 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2009/m/m5/b/ii/solution

For

j \geq 1

, the only incoming transition is from

j - 1

π_{j} = π_{j - 1} \frac{λ}{j} = π_{0} \frac{λ ^{j}}{j !} .

π_{0} e^{λ} = 1

, hence

π_{j} = e^{- λ} \frac{λ ^{j}}{j !}, j \geq 0 .

The zero-state balance also holds:

\sum_{i} π_{i} (1 - λ / (i + 1)) = 1 - \sum_{j \geq 1} π_{j} = π_{0}

. Thus the Poisson distribution is indeed stationary, including when

λ = 1

. Irreducibility gives uniqueness.

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