Let $X$ and $Y$ be continuous random variables with finite first and second moments. Then, is it true that $Var[X]\geq Var[E(X|Y)]$?



Is the unconditional variance of a RV an upper bound for the variance of any conditional expectation of the RV?
It is enough to show
$$
Var[X] - Var[E[X|Y]] \ge 0.
$$
By the fact $Var[X] = E[X^2] - E[X]^2$,
$$
Var[X] - Var[E[X|Y]] = E[X^2] - E[X]^2 - E\big[ E[X|Y]^2 \big] + E \big[E[X|Y]\big]^2 \\
= E\big[ X^2 - E[X | Y]^2 \big] \\
= E \big[ E\big[X^2 - E[X|Y]^2 \big| Y\big] \big] = E[Var[X|Y]\big] \ge 0,
$$
where the second and the third equality is due to the law of total expectation.


