Box-Muller Transformation: Polar Coordinates Interpretation

Question

I am aware that the Box-Muller transform leverages polar coordinates to arrive at the final transformations by plotting two uniform random variables, $(u, v)$ in the Cartesian plane. I have not seen a complete proof/explanation of why $\theta = 2\pi v$. Can someone here please provide a resource or an explanation of how to arrive at this result?

Answer 1

The idea of the Box-Muller transformation is that for a $N(0,I_2)$ random variable $X$, the PDF of $R=\sqrt{X_1^2+X_2^2}$ is $f(r)=re^{-r^2/2}$, while the angle is just uniformly distributed on $[0,2\pi)$ and independent of the radius, because of the rotational symmetry of the covariance matrix. (One way to prove this second statement carefully is to show that the distribution is preserved under rotations.)

Therefore you can generate $R$ according to that PDF and then use $(X_1,X_2)=(R\cos(\Theta),R\sin(\Theta))$ where $\Theta$ is uniformly distributed on $[0,2\pi)$. This is beneficial because the PDF for $R$, unlike the PDF for $X$, can be integrated and then inverted entirely in terms of elementary functions:

$$\int_0^x r e^{-r^2/2} dr = 1-e^{-x^2/2}$$

and so the probability integral transformation can be applied without the need for the inverse of the error function.