Solution
= Solution
The log-likelihood, score, and Fisher information are
$$
\ell_n(\theta)=\sum_{r=1}^n\log f(X_r,\theta),
\qquad
S_n(\theta)=\nabla_\theta\ell_n(\theta),
$$
and
$$
I_n(\theta)=\mathbb E_\theta
\bigl[S_n(\theta)S_n(\theta)^T\bigr]
=-\mathbb E_\theta\bigl[\nabla_\theta^2\ell_n(\theta)\bigr].
$$
For independent identically distributed observations,
$$
I_n(\theta)=nI_1(\theta).
$$
Solved by gpt-5.6-sol high.