= Solution
The <maximum likelihood estimation>[maximum-likelihood estimator] is a maximizer
$$
\widehat\theta\in\operatorname*{arg\,max}_{\theta}\ L_n(\theta),
\qquad L_n(\theta)=\prod_{i=1}^nf(X_i;\theta).
$$
Writing $\ell_n=\log L_n$ and the score $S_n=\nabla_\theta\ell_n$, the expected <Fisher information> <matrix> is
$$
\mathcal I_n(\theta)=\mathbb E_\theta[S_nS_n^T]
=-\mathbb E_\theta[\nabla_\theta^2\ell_n],
$$
where the equality uses the usual regularity conditions. For independent identically distributed observations, $\mathcal I_n=n\mathcal I_1$.
Under the standard identifiability, interior-parameter, <differentiability> and nonsingular-information assumptions, the large-sample distribution is
$$
\boxed{\sqrt n(\widehat\theta-\theta)\xrightarrow d
N(0,\mathcal I_1(\theta)^{-1}),}
$$
or approximately $\widehat\theta\sim N(\theta,\mathcal I_n(\theta)^{-1})$. Boundary parameters need not obey this regular approximation.
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