= Solution
Let $\ell_n(\theta)=\sum_{i=1}^n\log f(X_i;\theta)$ be the <log-likelihood>. Under the usual regularity conditions, the expected <Fisher information> in the whole sample is
$$
\boxed{I_n(\theta)=\mathbb E_\theta\!\left[\{\ell_n'(\theta)\}^2\right]
=-\mathbb E_\theta[\ell_n''(\theta)]=nI_1(\theta).}
$$
The equality of the two expressions uses <mean>-zero score and permitted differentiation of the normalization identity. Independent observations make the information additive. The <maximum-likelihood estimator> has <asymptotic normality>:
$$
\sqrt n(\widehat\theta-\theta)\xrightarrow{d}N(0,I_1(\theta)^{-1}),
\qquad
\boxed{\widehat\theta\ \text{approximately}\ N(\theta,I_n(\theta)^{-1}).}
$$
The factor $n$ is included in $I_n$; it must not be inserted a second time into the <variance>.
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