Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 31 / 4 / a / Solution

Use the gamma distribution shape-rate convention: $\Theta \sim \mathrm{Gamma}(\alpha ,\beta )$ with $\alpha ,\beta >0$. The conditional Poisson distribution has mean and variance both equal to $\Theta$, so the Bühlmann–Straub model parameters areFor $W={\sum }_j{m}_j$ and $y={\sum }_j{y}_j$, the Bühlmann–Straub credibility factor is $Z=W/(W+\beta )$. Thusand the required expected-count estimate isThe total exposure determines how much information the observed count carries; the number of years alone is not the appropriate denominator.