Box-Muller Transformation

A transformation which transforms from a two-dimensional continuous uniform distribution to a two-dimensional bivariate normal distribution (or complex normal distribution). If x_1 and x_2 are uniformly and independently distributed between 0 and 1, then z_1 and z_2 as defined below have a normal distribution with mean mu=0 and variance sigma^2=1.


This can be verified by solving for x_1 and x_2,


Taking the Jacobian yields

(partial(x_1,x_2))/(partial(z_1,z_2))=|(partialx_1)/(partialz_1) (partialx_1)/(partialz_2); (partialx_2)/(partialz_1) (partialx_2)/(partialz_2)|

See also

Bivariate Normal Distribution, Normal Deviate, Normal Distribution

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Box, G. E. P. and Muller, M. E. "A Note on the Generation of Random Normal Deviates." Ann. Math. Stat. 29, 610-611, 1958.

Referenced on Wolfram|Alpha

Box-Muller Transformation

Cite this as:

Weisstein, Eric W. "Box-Muller Transformation." From MathWorld--A Wolfram Web Resource.

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