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Two continuous random variables $X$ and $Y$ have joint distribution $p(x,y)$. How do I show that

  1. $E[X] = E_Y[E_X[X|Y]]$
  2. $var[X] = E_Y[var_X[X|Y]] + var_Y[E_X[X|Y]]$

I'm not sure how to set up the integrals here and what properties to use.

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    These formulas have some amazing geometric consideration via vector projections. I recommend to read it... – 2017-02-02
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    A first step would be to get rid of these horrible $E_Y$ and $E_X$, replacing each of them by $E$. A second step would be to consult a definition of $E(X\mid Y)$. – 2017-02-11

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For (1), we have that by definition of conditional expectations, we have \begin{align*} \int_B E_X(X \mid Y) dP = \int_B X dP \end{align*} for any $X \in \sigma(Y)$. Letting $B = \Omega$, we recover the tower property \begin{align*} \int_{\Omega} E_X(X \mid Y) dP &= \int_{\Omega} X dP \\ \Rightarrow E_Y(E_X(X \mid Y)) &= E(X). \end{align*} I will edit this post to comment on Axolotl's comment about showing (2) using vector projections when I have more time.