Let $X$ and $Y$ be \reftext{def:independence-events-rvs-2026a}{independent} \reftext{def:probability-space-random-variable-2026a}{random variables} on a probability space $(\Omega,\mathcal{F},P)$, each with finite \reftext{def:expectation-variance-2026a}{expectation}. Then the product $XY$ (a random variable, since $XY=\tfrac{1}{4}\bigl((X+Y)^2-(X-Y)^2\bigr)$ and sums, differences, and squares of random variables are random variables by Step 0(a) of the proof of \ref{thm:linearity-monotonicity-integral-2026a} and the power argument of \ref{def:expectation-variance-2026a}) has finite expectation, and

Consequently, if $X_1,\dots,X_r$ are independent random variables each having finite expectation and finite second moment, then for $i\ne j$,

and the \reftext{def:expectation-variance-2026a}{variance} is additive over independent summands: