Box-Muller transform

Independent uniform distribution draws $U_1,U_2$ on $(0,1)$ produce independent standard normal distribution draws $X_1=\sqrt{-2\log U_1}\cos (2\pi U_2)$ and $X_2=\sqrt{-2\log U_1}\sin (2\pi U_2)$. The proof transforms an independent uniform angle and radial density $r{e}^{-r^2/2}$ through the polar-coordinate Jacobian.