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?

  • $\begingroup$ I think you mean AC and BD in the first case ? (not CB) $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.