Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 2 / d / Solution

Let ${\ell }^{\ast }$ be a maximizing index for the original sample. Since $Z_i^{\prime }$ is at least the value of its ${\ell }^{\ast }$th linear form,and henceThe one-sided replacement form of the Efron–Stein inequality isFor independent uniform signs, $\mathbb{E}[(X_i-X_i^{\prime }{)}^2\mid X]=2$. It follows that