Box-Muller Transformation

The Box-Muller transform is a method for generating normally distributed random numbers from uniformly distributed random numbers. The Box-Muller transformation can be summarized as follows, suppose u₁ and u₂ are independent random variables that are uniformly distributed between 0 and 1 and let

then z₁ and z₂ are independent random variables with a standard normal distribution. Intuitively, the transformation maps each circle of points around the origin to another circle of points around the origin where larger outer circles are mapped to closely-spaced inner circles and inner circles to outer circles.
Let's see a Python snippet that implements the transformation:

from numpy import random, sqrt, log, sin, cos, pi
from pylab import show,hist,subplot,figure

# transformation function
def gaussian(u1,u2):
  z1 = sqrt(-2*log(u1))*cos(2*pi*u2)
  z2 = sqrt(-2*log(u1))*sin(2*pi*u2)
  return z1,z2

# uniformly distributed values between 0 and 1
u1 = random.rand(1000)
u2 = random.rand(1000)

# run the transformation
z1,z2 = gaussian(u1,u2)

# plotting the values before and after the transformation
figure()
subplot(221) # the first row of graphs
hist(u1)     # contains the histograms of u1 and u2 
subplot(222)
hist(u2)
subplot(223) # the second contains
hist(z1)     # the histograms of z1 and z2
subplot(224)
hist(z2)
show()

The result should be similar to the following:

In the first row of the graph we can see, respectively, the histograms of u₁ and u₂ before the transformation and in the second row we can see the values after the transformation, respectively z₁ and z₂. We can observe that the values before the transformation are distributed uniformly while the histograms of the values after the transformation have the typical Gaussian shape.