0

I have a question for which I have a somewhat unclear explanation.

Namely, if $X$ and $Y$ are each independent of $Z$ (i.e., $X$ is independent of $Z$ and $Y$ is independent of $Z$), then is $XY$ independent of $Z$?

My explanation, which is not entirely rigorous, is as follows:

$P(XY \in A, Z \in B) = \int_{\mathbb{R}} P(XY \in A, Z \in B | Y = y) dy = \int_{\mathbb{R}} P(XY \in A | Y = y) P(Z \in B | Y = y) dy = \int_{\mathbb{R}} P(XY \in A | Y = y) P(Z \in B) dy = P(XY \in A) P(Z \in B)$

Does this make sense?

  • 2
    Have you seen any examples of sets which are pairwise independent but not mutually independent? Here is a standard one if not. Try turning this into a counterexample: https://math.stackexchange.com/a/1783256/23078 – Chris Janjigian Apr 29 '24 at 19:09
  • https://math.stackexchange.com/questions/1739052/if-x-is-independent-to-y-and-z-does-it-imply-that-x-is-independent-to-yz?rq=1 – Michh Apr 29 '24 at 22:17

0 Answers0