 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-31/4/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Use the gamma distribution shape-rate convention:

Θ \sim Gamma (α, β)

with

α, β > 0

. The conditional Poisson distribution has mean and variance both equal to

Θ

m_{0} = v = \frac{α}{β}, a = \frac{α}{β ^{2}}, \frac{v}{a} = β .

For

W = \sum_{j} m_{j}

and

y = \sum_{j} y_{j}

Z = W / (W + β)

. Thus

μ = \frac{W}{W + β} \frac{y}{W} + \frac{β}{W + β} \frac{α}{β} = \frac{y + α}{W + β},

and the required expected-count estimate is

E [Y_{n + 1} ∣ Θ] = m_{n + 1} \frac{α + \sum _{j = 1}^{n} y _{j}}{β + \sum _{j = 1}^{n} m _{j}} .

The total exposure determines how much information the observed count carries; the number of years alone is not the appropriate denominator.

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