Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 27J / a

For every $a\in \mathbb{R}$,After squaring and taking expectations, the cross term vanishes because $\mathbb{E}[X-\mathbb{E}X]=0$. ThereforeThe second term is nonnegative and vanishes exactly at $a=\mathbb{E}X$, so the variance as the minimum mean squared error of a constant identity is