= Covariance of overlapping independent products
{title2=$\operatorname{Cov}(AB,BC)=\mathbb EA\,\mathbb EC\,\operatorname{Var}(B)$}
For mutually independent square-integrable <random variables> $A,B,C$, factor the joint moment as $\mathbb E[AB^2C]=\mathbb EA\,\mathbb E[B^2]\,\mathbb EC$, then subtract the product of the two means. This proves the formula. Sharing one factor can therefore produce nonzero <covariance> even when all original factors are independent. Products of disjoint groups of independent factors are themselves <independent random variables>.
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