For two random variables X and Y,
Prove that
$\displaystyle 1.E(X)=E[E(X/Y)]
2.V(X)=E(V(X/Y)]+V[E(X/Y)]$
This notation is unsatisfactory. You are using / rather than | to denote a conditional "thing".
You have a joint distribution $\displaystyle p(x,y)$ for you RV's, then:
$\displaystyle E(X)=\int \int x \;p(x,y)\; dy dx$
Also
$\displaystyle E(X|y)=\int x \; p(x|y) \;dx$
and $\displaystyle p(x|y)=\frac{p(x,y)}{p(y)}$
so:
$\displaystyle E(E(X|Y))=\int E(X|y) p(y) \; dy=\int \int x\; p(x,y)\;dx\;dy$
Now change the order of integration
CB