Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 28K / a / Solution

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.