I was looking through a friend's notes today and I saw an interesting exercise I wanted to try out. Unfortunately, I could not figure it out, so I turn to the internet.
"Given the model $$Y=\theta XU$$ where $U|X$ ~ $N(0,1)$ and $X$ & $\theta$ are strictly positive. What is the conditional expectation of Y given X?"
(It wasn't stated, but I assumed $\theta$ isn't a random variable but a constant. Not sure there).
My first instinct was that I need to do the Law of Iterated Expectations. The following is my rationale, so please let me know where I go wrong along the way.
Since we know $U|X$ ~ $N(0,1)$, we can say $E[U|X] = 0$. By Law of Iterated Expectations, this means $E[U] = 0$.
This next part is where I get tripped up. By law of Iterated Expectations, to get $E[Y|X]$, I'm not sure if the answer is $$E[E[Y|X]|X]$$ or $$E[E[Y|X]|U]$$
Either way, I get tripped up, and would appreciate insight in dealing with this problem. I haven't dealt too much with Iterated Expectations.