# Math Help - Bivariate distribution

1. ## Bivariate distribution

For two random variables X and Y,
Prove that
$1.E(X)=E[E(X/Y)]
2.V(X)=E(V(X/Y)]+V[E(X/Y)]$

2. Originally Posted by roshanhero
For two random variables X and Y,
Prove that
$1.E(X)=E[E(X/Y)]
2.V(X)=E(V(X/Y)]+V[E(X/Y)]$

This notation is unsatisfactory. You are using / rather than | to denote a conditional "thing".

You have a joint distribution $p(x,y)$ for you RV's, then:

$E(X)=\int \int x \;p(x,y)\; dy dx$

Also

$E(X|y)=\int x \; p(x|y) \;dx$

and $p(x|y)=\frac{p(x,y)}{p(y)}$

so:

$E(E(X|Y))=\int E(X|y) p(y) \; dy=\int \int x\; p(x,y)\;dx\;dy$

Now change the order of integration

CB