Past exam of the statistics course of the University of Oxford / 2023 / 3 / b / i / Solution

Writing $s={\sum }_i{x}_i$, the Poisson distribution likelihood is proportional to ${\theta }^s{e}^{-n\theta }$. Multiplying by the gamma distribution prior givesThis is Poisson-gamma conjugacy, with the rate convention made explicit:Its normalized density is $(\lambda +n{)}^{r+s}{\theta }^{r+s-1}{e}^{-(\lambda +n)\theta }/\Gamma (r+s)$ for $\theta >0$.