Can any one help me verify the following three equations? Do I make any mistake? If yes, can you give me some suggestions? Thank you in advance.
$\mathbb{E} \left ( X^{2} \cdot Var\left ( Y|X \right ) \right )=\mathbb{E} \left ( X^{2} \right ) \mathbb{E} \left ( Var\left ( Y|X \right ) \right ) $
$\mathbb{E} \left ( X^{2} \cdot \left [ \mathbb{E} \left ( Y|X \right ) \right ] ^{2} \right )=\mathbb{E} \left ( X^{2} \right ) \mathbb{E} \left ( \left [ \mathbb{E} \left ( Y|X \right ) \right ]^{2} \right )$
$Var\left ( \mathbb{E}\left ( XY|X \right ) \right ) = Var\left (X \cdot \mathbb{E}\left (Y|X \right ) \right )$