If $X, Z$ are independent, $Y, Z$ are independent, then are $XY,Z$ independent?
How about:
If $X,Y,Z$ are mutually independent, then are $XY, Z$ independent?
If none of the above is true, is there any such lemma that derives the independence from that of $X,Z$ and $Y,Z$?
Please show work or refer to a link, thank you!
For your first question, no: let $\epsilon_1$ and $\epsilon_2$ be two i.i.d. random variables taking the values $1$ and $-1$ with probability $1/2$. Let $X=\epsilon_1$, $Y=\epsilon_2$ and $Z=\epsilon_1\epsilon_2$.
For your second question, it is a consequence of the definition of mutual independence.