= Solution
Let
$$
\ell_n(\theta)=\log f_\theta(X_1,\ldots,X_n)
$$
be the <log-likelihood>. The <score function> and scalar <Fisher information> are
$$
S_n(\theta)=\frac{\partial}{\partial\theta}\ell_n(\theta),
\qquad
\boxed{I_n(\theta)=\mathbb E_\theta[S_n(\theta)^2]}.
$$
Under the usual regularity conditions the score has mean zero, so this is also its <variance>.
The information tensorizes when it is additive over observations. For identically distributed observations,
$$
\boxed{I_n(\theta)=nI_1(\theta)}.
$$
For an independent sample the joint score is the sum of the individual scores, and the <mean-zero score identity> makes all cross covariances vanish. This is the <Tensorization of Fisher information>.
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