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*