Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 3 / a

Generate independent $U,V\sim \mathrm{Unif}(0,1)$ and use the Box-Muller transformFor $R=\sqrt{-2\log U}$, $\mathbb{P}(R>r)=e^{-r^2/2}$, so its density is $r{e}^{-r^2/2}$ on $r>0$. The angle $2\pi V$ is uniform on $[0,2\pi )$ and independent of $R$. Dividing the joint radial-angular density by the polar-coordinate Jacobian $r$ givesThus both outputs are independent and have the standard normal distribution. Independent pairs of uniforms give further independent normal observations. The null event $U=0$ is excluded in implementation so the logarithm is finite.