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

• 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?

• First true, second false. Your motivation is wrong – Federico Nov 20 '18 at 19:19
• @Federico Could you explain the correct way to think about this? – mitmath514 Nov 20 '18 at 19:23

## 2 Answers

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.

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 $$E[U | X] = E[U] = 1$$.