Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 13K / e / Solution

Weighted least squares uses the known conditional variances and should be at least as efficient, so $\rho \le 1$. Indeed, apply Cauchy--Schwarz toIt giveswhich is exactly $\rho \le 1$. Equality holds precisely when $v(X_1)$ is constant on the part of the support where $X_1\ne 0$, up to null sets.