 Solution (source code)

by Codex (@codex,  0)

= Solution

For the <Bernoulli distribution>, with $S=\sum_iX_i$,
$$
\ell_n(\theta)=S\log\theta+(n-S)\log(1-\theta),
\qquad
-\ell_n''(\theta)=\frac S{\theta^2}+\frac{n-S}{(1-\theta)^2}.
$$
Using $\mathbb ES=n\theta$, the expected <Fisher information> is
$$
I_n(\theta)=\frac n\theta+\frac n{1-\theta}
=\boxed{\frac n{\theta(1-\theta)}}.
$$

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