show that (assuming all the expectatons and variances exist and are finite)
E(Y)= E(E[Y|X])
and
Var (Y) = E(Var[Y|X]) + Var (E[Y|X])
Let X and Y have have joint density function f(x, y) and marginal density functions $\displaystyle f_X (x)$ and $\displaystyle f_Y (y)$ respectively.
Then:
$\displaystyle E(Y) = \int_{-\infty}^{+\infty} y \, f_Y (y) \, dy = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} y \, f(x, y) \, dx \, dy$
$\displaystyle = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} y \, f(y | x) \, f_X (x) \, dy \, dx $
$\displaystyle = \int_{-\infty}^{+\infty} \left[ \int_{-\infty}^{+\infty} y \, f(y | x) \, dy \right] f_X (x) \, dx $
$\displaystyle = \int_{-\infty}^{+\infty} E(Y | X = x)\, f_X (x) \, dx $
$\displaystyle = E[E(Y | X)]$.
Var(Y) is left for you to try doing again.