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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.

z_1=sqrt(-2lnx_1)cos(2pix_2)
(1)
z_2=sqrt(-2lnx_1)sin(2pix_2).
(2)

This can be verified by solving for x_1 and x_2,

x_1=e^(-(z_1^2+z_2^2)/2)
(3)
x_2=1/(2pi)tan^(-1)((z_2)/(z_1)).
(4)

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)|
(5)
=-[1/(sqrt(2pi))e^(-z_1^2/2)][1/(sqrt(2pi))e^(-z_2^2/2)].
(6)

See also

Bivariate Normal Distribution, Normal Distribution

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References

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. https://mathworld.wolfram.com/Box-MullerTransformation.html

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