Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 18H / a

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- Solution a

Solution

Let

d_{t} = # {i : X_{i} = t}, r_{t} = # {i : X_{i} \geq t} = n - \sum_{s < t} d_{s} .

Thus

d_{t}

patients recover during month

t

, while

r_{t} - d_{t}

survive that month without recovery. The likelihood function is

L (q_{1}, \dots, q_{T}) = t = 1 \prod T q_{t}^{d_{t}} (1 - q_{t})^{r_{t} - d_{t}} .

The independent

Beta (T, 1)

prior densities are proportional to

q_{t}^{T - 1}

. Beta-binomial conjugacy therefore leaves the coordinates posteriorly independent, with

q_{t} ∣ X \sim Beta (T + d_{t}, 1 + r_{t} - d_{t}) .

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