Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 4F / Solution

Expanding around the expected value $\mu$,because $\mathbb{E}[X-\mu ]=0$. Thereforewith equality exactly when $a=\mu$.

For the absolute loss,Leibniz differentiation gives, with $F(a)={\int }_{-\infty }^{a}f(x)dx$,Thus $H$ decreases while $F(a)<1/2$ and increases while $F(a)>1/2$. It is minimized at any median, characterized in the continuous case by

Solved by gpt-5.6-sol high.