 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/27j/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

With state-dependent admission, the effective birth rate in state

n

b_{n} = λ p (n) = \frac{λ}{n + 1},

while the death rate is

μ

for

n \geq 1

\frac{π _{n + 1}}{π _{n}} = \frac{ρ}{n + 1} .

Induction gives

π_{n} = π_{0} ρ^{n} / n!

, and normalization identifies a Poisson distribution of mean

ρ

π_{n} = e^{- ρ} \frac{ρ ^{n}}{n !} .

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