# Are the products of different independent random variables independent?

I'm confused about the independence of the product of independent random variables.

Let A and B be independent of each other and C and D also be independent of each other, then I understand that AC and CB are not necessarily independent each other.

However, A, B, C, and D are mutually independent of each other.

In that case, are AC and BD independent of each other?

• I think you mean AC and BD in the first case ? (not CB) – Stéphane Laurent Aug 2 '12 at 7:41

The answer is: yes. If $A$, $B$, $C$ and $D$ are mutually independent, that means that:

• $A$ si independent of $(B,C,D)$

• $B$ is independent of $(A,C,D)$

• ....

• $(A,B)$ is independent of $(C,D)$

• $(A,C)$ is independent of $(B,D)$

• ...

Moreover, saying that $(A,C)$ is independent of $(B,D)$ means that for all (Borelian) real-valued functions $f$ and $g$, the two random variables $f(A,C)$ and $g(B,D)$ are independent, thereby implying the answer to your question.