Box–Muller transform
The Box–Muller transform, by George Edward Pelham Box and Mervin Edgar Muller 1958, is a pseudo-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.
It 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 an even more efficient method. Furthermore, the Box–Muller transform can be employed for drawing from truncated bivariate Gaussian densities.
Suppose U1 and U2 are independent random variables that are uniformly distributed in the interval (0, 1). Let
and
Then Z0 and Z1 are independent random variables with a standard normal distribution.
The derivation is based on a property of a two-dimensional Cartesian system, where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for R2 and Θ (shown above) in the corresponding polar coordinates are also independent and can be expressed as
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