# 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?

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.