Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 4 / i / Solution

A beta distribution is a convenient prior for a probability $p\in (0,1)$ and is conjugate to the binomial distribution of the death count. Match its mean $m=0.05$ and variance $v=0.02^2=0.0004$. For $p\sim \mathrm{Beta}(\alpha ,\beta )$,The moment matching for a beta prior formula givesConditional on $p$, use $D\sim \mathrm{Binomial}(90,p)$ and the observed count $D=9$. Multiplying its likelihood $p^9(1-p{)}^{81}$ by the prior density gives the Bayesian posterior through Beta-binomial conjugacy:Under squared-error loss, take its posterior mean as the point estimate. A 95% equal-tail credible interval is given by its 0.025 and 0.975 quantiles. Numerically,Matching two moments does not uniquely determine a prior distribution; the beta family is a justified convenient choice, not a conclusion forced by those moments. Its transfer to this hospital presumes that the historical between-hospital distribution is relevant to the present risk and case mix.