I am slightly confused about the different between
$\mathrm{E}[Y|X = x]$
and
$\mathrm{E}[Y|X]$
and similarly for Variance.
It seems to me the first should be a scalar, because we first pick a specific $X = x$ and then get the expected value of $Y$ within that set whereas the second one is a random variable that depends on the random variable $X$. Is that correct?
Any definition using the probabilities $\mathrm{P}(X)$, $\mathrm{P}(Y)$, $\mathrm{P}(Y|X)$ and $\mathrm{P}(Y, X)$ is appreciated.