Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 11F / b

For real random variables with finite second moments, the Cauchy-Schwarz inequality isThe product is integrable because $2|Y_1{Y}_2|\le Y_1^2+Y_2^2$. If $\mathbb{E}[Y_2^2]=0$, then $Y_2=0$ almost surely and the result is immediate. Otherwise the nonnegative expectation of a square gives, for every real $t$,Choose $t=\mathbb{E}[Y_1{Y}_2]/\mathbb{E}[Y_2^2]$. The resulting inequality is exactly the claimed Cauchy-Schwarz inequality. Equality holds precisely when $Y_1$ is a constant multiple of $Y_2$ almost surely in this nondegenerate case.