= Solution
Let $S=\sum_i x_i$ and $\bar x=S/n$. Terms independent of $p$ can be dropped from the <log-likelihood>, leaving
$$
\ell(p)=nr\log p+S\log(1-p)+C,
\quad\ell'(p)=\frac{nr}p-\frac S{1-p},
\quad\ell''(p)=-\frac{nr}{p^2}-\frac S{(1-p)^2}.
$$
For $S>0$, strict concavity gives the unique interior <maximum-likelihood estimator>[maximum likelihood estimator]
$$
\boxed{\widehat p=\frac{nr}{nr+S}=\frac r{r+\bar x}}.
$$
If $S=0$, 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> is
$$
I_n(p)=-\mathbb E\ell''(p)
=\frac{nr}{p^2}+\frac{nr(1-p)/p}{(1-p)^2}
=\boxed{\frac{nr}{p^2(1-p)}}.
$$
The information per observation is $I_1(p)=r/[p^2(1-p)]$.
Back to article page
