I See this identity but it does not make sense to me.
$ Var(X)= EVar(X|Y)+Var(E(X|Y)) $
is this something that should intuitively make sense? Is it saying the variance of X is equal to the expected variance of X given Y plus the variance of Y? I say the second part because since E(X|Y) is a function with Y as variable Var(E(X|Y)) would capture the variance of Y.