I'm trying to show that if $\displaystyle \rho = 0$ on the bivariate normal, X and Y are independent. My approach is to show $\displaystyle f(x, y)=f_{X}(x) f_{Y}(y)$

Then $\displaystyle f_{X}(x) = \int_{-\infty}^{\infty}{f(x, y) \ dy}$ and same for $\displaystyle f_{Y}(y)$

Can someone tell me if this approach works? Thanks!