 # Box Muller Transform: Simple Definition

Share on

A Box Muller transform takes a continuous, two dimensional uniform distribution and transforms it to a normal distribution.

It is widely used in statistical sampling, and is an easy to run, elegant way to come up with a standard normal model. In fact, since it can be used to generate normally distributed random numbers, it was originally developed as a better and computationally efficient alternative to inverse sampling.

## Running the Box Muller Transformation

At its most basic, the Box-Muller transformations simply takes two variables that are uniformly distributed and sends them to two independent random variables with a standard normal distribution.

Let’s say U1 and U2 are our original independent random variables; they are uniformly distributed in the interval (0,1). The Box-Muller transformation creates new Z0 and Z1; independent, random variables that have a standard normal distribution:  The derivation here is based on the way we can represent any point in the X,Y Cartesian plane through polar coordinates, with a radius and an angle.

## The Polar Form of the Box Muller Transform

Although the same points are being mapped to as with the ‘basic form’ of the Box-Muller transform, given above, this alternate form is sometimes preferred because it replaces sine and cosine with simple divisions and so is often more computationally efficient.

## References

Goodman, Jonathan. Lecture Notes on Monte Carlo Methods. Chapter 2: Simple Sampling of Gaussians Retrieved from https://www.math.nyu.edu/faculty/goodman/teaching/MonteCarlo2005/notes/GaussianSampling.pdf on March 16, 2018

Vafa, Keyon. The Box Muller Transform.