Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 27K / c / Solution

Put ${\mu }_f={\int }_0^{1}fd\lambda$. Subtracting this constant does not affect variance, and pairwise independent random variables have zero pairwise covariance. ThereforeOnly pairwise independence is needed for the off-diagonal covariance terms to vanish.