Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 3 / c

Table of contents
- Solution c

Solution

Averaging the slow propensities over the conditional Poisson distribution uses

E [Y (Y - 1) ∣ X = x] = q (x)^{2}

. Thus the effective birth and death rates of

X

are

λ_{1} (x) = \frac{α _{1}}{V} q (x)^{2} = \frac{α _{1} α _{3}^{2} V ^{3}}{α _{4}^{2} x ^{2}},

λ_{2} (x) = \frac{α _{2}}{V} x (x - 1) .

Consequently

\partial_{t} p_{0} (x, t) = ([E_{x}^{- 1} - 1] λ_{1} (x) + [E_{x}^{+ 1} - 1] λ_{2} (x)) p_{0} (x, t) .

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