Box-Muller Transformation

Definition

Let $U_1,U_2\stackrel{iid}{\sim }U(0,1)$, $R^2/2\sim Exp(1)$, and $\theta \sim U(0,2\pi )$, then $X_1=R\cos (\theta )=\sqrt{-2log{U}_1}\cos (2\pi U_2)\sim \mathrm{N}(0,1)$ and $X_2=R\sin (\theta )=\sqrt{-2log{U}_1}\sin (2\pi U_2)\sim \mathrm{N}(0,1)$

The method that generates pairs of i.i.d. normal distribution by using two uniform distributions or Exponential Distribution