Box–Muller transform

Visualisation of the Box–Muller transform — the coloured points in the unit square (u₁, u₂), drawn as circles, are mapped to a 2D Gaussian (z₀, z₁), drawn as crosses. The plots at the margins are the probability distribution functions of z0 and z1. z0 and z1 are unbounded; they appear to be in [−2.5, 2.5] due to the choice of the illustrated points. In the SVG file, hover over a point to highlight it and its corresponding point.

In mathematics, the Box–Muller transform, introduced by George Edward Pelham Box and Mervin Edgar Muller, is a random number sampling method for generating pairs of independent, standard, normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers. The method was first mentioned explicitly by Raymond E. A. C. Paley and Norbert Wiener in their 1934 treatise on Fourier transforms in the complex domain. Given the status of these latter authors and the widespread availability and use of their treatise, it is almost certain that Box and Muller were well aware of its contents.

The Box–Muller transform is commonly expressed in two forms. The basic form as given by Box and Muller takes two samples from the uniform distribution on the interval $(0,1)$ and maps them to two standard, normally distributed samples. The polar form takes two samples from a different interval, $[-1,1]$, and maps them to two normally distributed samples without the use of sine or cosine functions.

The Box–Muller transform was developed as a more computationally efficient alternative to the inverse transform sampling method. The ziggurat algorithm gives a more efficient method for scalar processors (e.g. old CPUs), while the Box–Muller transform is superior for processors with vector units (e.g. GPUs or modern CPUs).