Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 2 / d / Solution

Subtract a constant so that $f(0)=0$, and write $m=\mathbb{E}f(Y)$, $V=\mathrm{Var}f(Y)$, and $A=\mathbb{E}{f}^{\prime }(Y{)}^2$. The supplied identity applied to $f$ and $f^2$ givesBy Cauchy-Schwarz inequality, $m^2\le A$ andThus $V+m^2\le 2\sqrt{(V+m^2)A}$, so $V+m^2\le 4A$ and in particular $V\le 4A$. Therefore the standard Laplace distribution has $C_P(Y)\le 4$.