Past exam of the options I course of the University of Oxford / 2009 / O / O2 / i / Solution

Let $S={\sum }_i{x}_i$ and $\bar{x}=S/n$. Terms independent of $p$ can be dropped from the log-likelihood, leavingFor $S>0$, strict concavity gives the unique interior maximum likelihood estimatorIf $S=0$, the likelihood increases up to $p=1$: this is the boundary estimate on the closed parameter range, or a supremum if the model insists on $p<1$.

Using the given mean, the sample Fisher information isThe information per observation is $I_1(p)=r/[p^2(1-p)]$.