TITLE: $ \mathbb{E}\left[\big(\mathbb{E}[X|Y]\big)^2\right] \leq \mathbb{E}(X^2)$
QUESTION [0 upvotes]: I'm working on old qualifying exam problems and haven't been able to get this one yet.

Let $X,Y$ be random variables with joint density function $f_{XY}$, $\mathbb{E}(X^2)<\infty$, and $\mathbb{E}(Y^2)<\infty$.
Show that $ \mathbb{E}\left[\big(\mathbb{E}[X|Y]\big)^2\right] \leq \mathbb{E}(X^2). $

I assume using Jensen's inequality and the law of total expectation will help. I tried using those and also just applying the definition of conditional expectation, but I haven't gotten it.
Any suggestions or hints would be greatly appreciated!

REPLY [3 votes]: As an alternative to @Byron's excellent suggestion, expand the inequality
$$
E[(X-E[X\mid Y])^2]\geqslant0,
$$
using the fact that
$$
E[XE[X\mid Y]]=E[E[X\mid Y]^2].
$$
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