Expectation of a product of three random variables with partial independence

Question

I'm struggling with the following question:

Suppose we have three random variables: $X,Y,Z$ and suppose that $X$ and $Z$ are independent and $Y$ and $Z$ are independent. Is it true that $$\mathbb{E}[XYZ]=\mathbb{E}[XY]\mathbb{E}[Z]?$$

It looks to me as something that is probably not true. I don't see a reason why $Z$ should be independent of the $\sigma$-algebra generated by both $X$ and $Y$ but I'm struggling to find a specific counterexample for the problem above.

Answer 1

No. There exist events $A,B,C$ such that any two of them are independent but $P(A\cap B\cap C) \neq P(A)P(B)P(C)$. If you take $X=I_A,Y=I_B,Z=I_C$ you will get a counterexample. Example: toss two fair coins independently and consider thwe following events:

a) the outcomes are both heads or both tails

b) the first one is heads

c) the second one is heads.

You can check that these events are pairwise independent but not jointly independent.

Answer 2

Let $X,Y$ be iid uniform $\pm1$ random variables, let $Z=XY$. Then $XYZ=1$ with probability 1 and $X$, $Y$, and $Z$ are pairwise independent. Since $1=E[XYZ]$ and $E[XY]=E[Z]=0$, you have your counterexample.