The Cauchy-Schwarz Root of Probability’s Hidden Clarity
In probability theory, the Cauchy-Schwarz inequality stands as a cornerstone of mathematical clarity, revealing deep structure in randomness through a deceptively simple bound: for independent random variables \(X\) and \(Y\), $$(\mathbb{E}[XY])^2 \leq \mathbb{E}[X^2]\mathbb{E}[Y^2].$$ This inequality does more than bound covariance—it identifies the Cauchy-Schwarz root, the minimal variance configuration under independence, a foundational threshold for trustworthy […]