Let A and B be two random variables, both independent from another random variable C. Is A*B also independent from C?

Let A and B be two random variables both independent from another random variable C. If A is independent from B, is A*B also independent from C? And if A and B are no independent from each other?

Answer

If all you have is pairwise independence then there is a counterexample. Suppose the following four cases each have probability $\frac14$:

A B C  AB

0 0 0  0 

0 1 1  0 

1 0 1  0 

1 1 0  1

Then $A$ is independent of $C$ and $B$ is independent of $C$, and $A$ is independent of $B$.

But $AB$ and $C$ are not independent as $\mathbb P(AB=1\mid C=0)=\frac12\not= 0=\mathbb P(AB=1 \mid C=1)$

In this example $A$, $B$ and $C$ are pairwise independent as suggested by the question, but are not mutually independent. If they had been mutually independent then it would also follow that $AB$ would be independent of $C$. A slightly weaker condition is that if $A$ and $B$ were jointly independent of $C$ then it would follow that $AB$ would be independent of $C$ even if $A$ and $B$ were not independent of each other.

Attribution
Source : Link , Question Author : Fortià Vila , Answer Author : Henry

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