# Box–Muller transform

## Statistical transform / From Wikipedia, the free encyclopedia

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The **Box–Muller transform**, by George Edward Pelham Box and Mervin Edgar Muller,[1] 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 in fact first mentioned explicitly by Raymond E. A. C. Paley and Norbert Wiener in 1934.[2]

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.[3] 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).[4]