Definition of $E(X|Y=y)$ and $E(X|Y)$. Why do they differ?

Question

I know the mathematical formulas for both $E(X|Y=y)$ and $E(X|Y)$ but I was just wondering if someone could describe to me just in plain words the meaning of these symbols and how they differ from one another.

I know they are calculating conditional expectation (value of X given Y) but I'm looking for a more descriptive definition to help me grasp the concepts. I think $E(X|Y)$ is still a random variable and $E(X|Y=y)$ is an actual number.

Answer 1

$E[X|Y=y]$ is just a numerical function of $y$, it's not random. By contrast, $E[X|Y]$ is a random variable (though it can potentially be constant, even then it should be thought of as a random variable).

The interesting but intuitive fact is that when you define $E[X|Y]$ as $E[X|\sigma(Y)]$, you have $E[X|Y]=E[X|Y=y]$ when $Y=y$. In other words, $E[X|Y]$ is a function only of $Y$. So these two things are very closely related.

Answer 2

$E(X|Y)$ is a function of the random variable $Y$.

$E(X|Y=y)$ is a function of $y$.

$g(Y) = E(X|Y)$

$h(y) = E(X|Y=y)$