Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 17H / c / Solution

PutIndependence and $\mathrm{var}(Y_i)=\theta x_i$ givewhereas ${\sum }_i{Y}_i\sim \mathrm{Poisson}(\theta S_1)$ givesThe Cauchy-Schwarz inequality, applied to $(x_i^{1/2})$ and $(x_i^{3/2})$, yieldsConsequentlyEquality holds exactly when all positive exposures $x_i$ are equal. This is the variance comparison for Poisson exposure estimators.