# Box-Muller Transform Normality

I'm trying to show that for uniformly distributed variables $X_1$ and $X_2$ that the vector $$(\sqrt{-2\log X_1} \cos(2\pi X_2), \sqrt{-2 \log X_1} \sin(2\pi X_2))$$ is two-dimensional normal. Letting $Y_1$ be the first component and $Y_2$ be the second component I see that $X_1 = e^{-(Y_1^2 +Y_2^2)/2}$ and $X_2 = \frac{1}{2\pi} \arctan(Y_2/Y_1)$. Is there a way to proceed from here to show that $(Y_1, Y_2)$ is normal?

• Hint: do a transformation from cartesian to polar coordinates. Nov 14 '14 at 18:49