A transformation which transforms from a two-dimensional continuous uniform distribution to a two-dimensional bivariate
normal distribution (or complex normal
distribution). If 
and 
are uniformly and independently distributed between 0 and
1, then 
and 
as defined below have a normal distribution
with mean 
and variance 
.
 |  |  |
(1)
|
 |  |  |
(2)
|
This can be verified by solving for 
and 
,
 |  |  |
(3)
|
 |  |  |
(4)
|
Taking the Jacobian yields
 |  |  |
(5)
|
 |  | ![-[1/(sqrt(2pi))e^(-z_1^2/2)][1/(sqrt(2pi))e^(-z_2^2/2)].](/images/equations/Box-MullerTransformation/Inline26.svg) |
(6)
|
