I may have missed this during my intro stats/prob course but what is the difference between:
$E_Y[X]$ and $E[X|Y]$?
It seems like one you are marginalizating over and the other you are conditioning on. I keep seeing the following:
$E[X] = E_Y[E[X|Y]]$. I do not understand, intuitively, why this is true. Can someone please explain this to me?
Here is some intuition. In the inner expectation, you have $E(X\mid Y)$, which is a function of $Y$. You are conditioning on $Y$, and then you integrate out $X$, leaving you an expression dependent only on $Y$. Then you take the expected value with respect to the distribution on $Y$ to recover the original expected value, which no longer depends on the random variables $X$ and $Y$.