Are the products of dependent and independent random variables independent? Let A, B, C, and D be four random variables such that A and B are independent, and C and D are dependent. It is unknown whether A and C are independent nor whether B and D are independent. Let E and F represent the products E = AC and F = BD. Are E and F necessarily independent?
If not, say we add the knowledge that A and C are independent and B and D are independent; now are E and F necessarily independent?
 A: First, define dependent to mean not independent, that is, the joint distribution is not the product of the marginal distributions. Note also that all constant variables are independent of everything. 
Though this may look like cheating, if $A = B = 1$ and $C = D \in \{0,1\}$, with the constraint that their common distribution is not degenerate, then $A$ and $B$ are independent, $C$ and $D$ are not, and since $E = C$ and $F = D$, then $E$ and $F$ are not independent either. Furthermore, $A$ and $C$ are independent and $B$ and $D$ are independent by degeneracy of the distributions of $A$ and $B$.
A: In both cases, the answer is No, $E = AC$ and $F = BD$ are not necessarily independent.  Your "added knowledge" makes $A, B, C, D$ pairwise independent but that is not sufficient to guarantee that $AC$ and $BD$ are mutually independent.  If $A, B, C, D$ are _mutually independent, then $AC$ and $BD$ are mutually independent events.
A: Given that C and D are dependent, E and F are definitely NOT independent. Rather they are dependent as well. 
Even if A & C is independent, and B & D is independent, the above dependency will remain same.
