Box-Muller transform

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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)].