= Variance identity for independent copies
{title2=$\frac12\mathbb E|Q-Q'|^2=\mathbb E|Q|^2-|\mathbb EQ|^2$}
For <independent> identically distributed <random vectors> $Q,Q'$ with finite second <absolute moments>, expand $|Q-Q'|^2=|Q|^2+|Q'|^2-2Q\cdot Q'$. Their <independence> makes $\mathbb E(Q\cdot Q')=|\mathbb EQ|^2$, proving the identity. The right side is the sum of the coordinate <variances>. Applied to the displacement in an optimal <transport plan>, it separates motion of the <barycenter of a measure> from relative particle motion.
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