Could you help me to show the following:
"Consider the random variables $X,Y$ i.i.d. uniform on $[-1,1]$. Then $(X,Y,XY)$ are pairwise independent but mutually dependent. "
I am confused about pairwise independent: how can $X,XY$ be pairwise independent?
I would also appreciate some more general clarification on pairwise independence, mutual independence and higher order interactions: I am reading here, page 1:
"Tests of total (mutual) and pairwise independence are insufficient, however, since they do not rule out all third order factorizations of the joint distribution. An important class of high order interactions occurs when the simultaneous effect of two variables on a third may not be additive. In particular, it may be possible that $X\perp Y$ and $Y\perp Z$, whereas $(X,Y)\perp Z$ does not hold.''
the adjective "third" in the expression "third order factorizations" comes from the fact that I am considering 3 variables? So, if I have 4 variables then I have also fourth orderfactorizations? What is exactly intended as third (or higher) order factorizations?
What does it mean that the effect of a variable on another is additive? Is the example above reporting an additive effect? It seems multiplicative.
"in particular, it may be possible that $X\perp Y$ and $Y\perp Z$, whereas $(X,Y)\perp Z$ does not hold." Isn/t this case captured as well by the mutual dependence situations mentioned above?