Re: Expected Value Question

Hello,

I suggest you write the expectations with respect to which rv you consider it : $\displaystyle E_X[m(X)]=\int m(x) ~dP_X$ for example.

So we actually have $\displaystyle g(x)=E_Y[f(x,Y)]$, we want to prove that $\displaystyle E_X[g(X)]=E_{(X,Y)}[f(X,Y)]$

But $\displaystyle E_X[g(X)]=E_X[E_Y[f(X,Y]]=E_X\left[\int_\Omega f(X,y) ~dP_Y\right]=\int_\Omega \int_\Omega f(x,y)~dP_Y dP_X$

And we also have $\displaystyle E_{(X,Y)}[f(X,Y)]=\int_{\Omega^2} f(x,y)~dP_{(X,Y)}$

But since X and Y are independent, $\displaystyle dP_{(X,Y)}=dP_X dP_Y$

So $\displaystyle E_{(X,Y)}[f(X,Y)]=\int_\Omega \int_\Omega f(x,y) ~dP_Y dP_X$

(I think this is where the positiveness of f intervenes, in order to apply Fubini's theorem)

And hence the equality.