Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 2 / a / Solution

Take the exposure $E_i$ as known and positive. The Poisson distribution log-likelihood, up to a parameter-independent constant, is ${\ell }_i({\lambda }_i)=y_i\log {\lambda }_i-E_i{\lambda }_i$. Its first two derivatives areSince $\mathbb{E}(Y_i)=E_i{\lambda }_i$, the Fisher information isThe Jeffreys prior is proportional to its square root. Removing the constant $\sqrt{E_i}$ givesThis is an improper prior: its integral diverges at infinity.