Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 4F / a / Solution

Put $U=X-EX$ and $V=Y-EY$. Their second moments are the finite positive variances of $X,Y$. The Cauchy-Schwarz inequality for square-integrable random variables givesDivide by the positive product of variances and take square roots. The correlation bound is $\boxed{-1\le \rho (X,Y)\le 1}$. As usual, non-constant here means not almost surely constant, so that the denominator is nonzero.