Does conditional expectation imply anything about the expected value of the product of two random variables?

Question

• If $E[U|X]=0$ then $[XU] = 0$

• If $E[XU]=0$ then $[U|X] = 0$

Which of the two statements above are true? This is my thought process for the first one:

if $E[U|X]=0$ then $E[U]=0$ and if U and X are independent, then $E[XU]=E[X]E[U]=E[X]*0=0$

which means the first statement is true if X and U are independent.

Is this the right way to think about it? What about the second one?

Answer 1

For the first statement you should apply the Law of Iterated Expectations.

If $E[U|X]=0$ then $E[UX]=E[E[UX|X]]=E[X\cdot E[U|X]]=E[X\cdot0]=0$

The second statement doesn't hold.

Answer 2

To see why the first statement is true, I would follow Ramiro Scorolli's excellent answer and use the Law of Iterated Expectations: $E[UX]=E[E[UX|X]]=E[X\cdot E[U|X]]$, so $E[U|X]] = 0$ implies that $E[XU]=0$. For the second statement, a very simple counterexample would be to take $X = 0$ and $U = 1$ with probability one; i.e. $X$ and $U$ are constant random variables. Then $E[XU] = E[0 \cdot 1] = 0$, but $E[U | X] = E[U] = 1$.