If $c_i=a_ib_i, c_j=a_jb_j,$ $a_i$ and $a_j$ are independent, $b_i$ and $b_j$ are dependent, what about $c_i$ and $c_j$? Are $c_i$ and $c_j$ independent? ($A, B, C$ are r.v.s., $A$ and $B$ are independent as well)
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1depends on if $a_i$ and $b_j$ (or $a_j$ and $b_i$) are independent or not :-) – jcken Sep 21 '20 at 08:22
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Sorry, $A$ and $B$ are independent and $\text{Cov}(b_i, b_j) \neq 0.$ – Sep 21 '20 at 08:24
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The map $(A,B)\to AB=C$ is a function of the random variable $(A,B).$ That's why the duplicate applies. – whuber Sep 21 '20 at 13:27