Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 26K / b / Solution

The per-observation Fisher information is $I(\theta )={\mathbb{E}}_{\theta }[({\partial }_{\theta }\log f(X_1,\theta ){)}^2]$. Sinceand the integral of $f_{\theta \theta }$ is the second derivative of $1$, the regularity assumptions giveFor independent observations the total Fisher information is $nI(\theta )$, because centered individual score functions have zero cross terms.