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For one observation with density $f(x,\theta )$, the Fisher information matrix at ${\theta }_0$ isHereSince ${\mathrm{Cov}}_{{\theta }_0}(X)=I_p$, the Fisher information of a multivariate normal location model isThe information in the full independent sample is $n{I}_p$ by Tensorization of Fisher information.

For observations $x=(x_1,\dots ,x_n)$, the likelihood function isA maximum-likelihood estimator is any measurable choiceFor one observation in a regular scalar model, the score function isand the Fisher information iswhere the second identity requires the usual differentiation-under-the-integral regularity. For $n$ independent observations, Tensorization of Fisher information gives $I_n(\theta )=nI(\theta )$.

Letbe the log-likelihood. The score function and scalar Fisher information areUnder 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,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.