If I have four random variables $X_1, X_2, Y_1, Y_2$ such that $X_i \perp Yj$ for $i,j \in \left\{{1,2}\right\}$, $X_1 \perp X_2$, $Y_1 \perp Y_2$, it could be say that $\mathbb{E}[X_1X_2Y_1Y_2]=\mathbb{E}[X_1]\mathbb{E}[X_2]\mathbb{E}[Y_1]\mathbb{E}[Y_2]$
2026-05-15 19:58:03.1778875083
Independence of components of random vectors
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No, sadly. You need joint independence, not only pairwise.
Take $X_1,X_2,Y_1$ to be (jointly) independent Rademacher random variables (i.e., uniform on $\{-1,1\}$), and set $Y_2=X_1X_2Y_1$. Then $X_1,X_2,Y_1,Y_2$ satisfy the assumptions, but...