The Box-Muller Transform

Computers generate uniform random numbers easily (any value equally likely between 0 and 1), but simulations — pricing an option, running a Monte Carlo risk model — need normally distributed random numbers instead. The normal distribution's cumulative distribution function has no simple closed-form inverse, so directly transforming a uniform draw into a normal one isn't straightforward. The Box-Muller transform sidesteps this by using two uniform draws at once and a bit of trigonometry to produce two independent standard normal draws exactly, with no approximation.

Given two independent uniform random numbers $U_1, U_2$ on $(0,1)$, the transform computes

$$Z_1 = \sqrt{-2\ln U_1}\,\cos(2\pi U_2), \qquad Z_2 = \sqrt{-2\ln U_1}\,\sin(2\pi U_2),$$

and $Z_1, Z_2$ are exactly independent, standard normal random variables. The $\sqrt{-2\ln U_1}$ term generates a random radius and the $2\pi U_2$ term generates a random angle, so geometrically the pair $(Z_1, Z_2)$ is just a uniformly random angle at a radius drawn from the right distribution — which works out, after the math, to be exactly bivariate normal.

A worked example

Draw $U_1 = 0.5$ and $U_2 = 0.25$. Then $\sqrt{-2\ln 0.5} = \sqrt{1.386} \approx 1.177$, and $2\pi \times 0.25 = \pi/2$, so $\cos(\pi/2) = 0$ and $\sin(\pi/2) = 1$. This gives $Z_1 = 1.177 \times 0 = 0$ and $Z_2 = 1.177 \times 1 = 1.177$ — two standard normal draws obtained from nothing more than two uniform numbers and a square root, log, and pair of trig functions.