Box Muller Transformation

TL;DR

• Converts two uniform(0,1) random numbers into two normally distributed random numbers.
• Simple to implement and commonly used in statistical and mathematical applications where normal samples are needed.
• Works with uniformly generated inputs (e.g., from a linear congruential generator) to produce outputs with mean 0 and standard deviation 1 in the example shown.

Definition

The Box-Muller Transformation is a method used to generate normally distributed random numbers by transforming two uniformly distributed random numbers into two normally distributed random numbers.

The transformation uses the following equations:

$$z_1 = \sqrt{-2\ln(x)}\cos(2\pi y)$$ $$z_2 = \sqrt{-2\ln(x)}\sin(2\pi y)$$

where ln is the natural logarithm and π is the mathematical constant.

Explanation

• Start with two uniformly distributed random numbers, typically in the interval (0, 1). These can be produced by a uniform random number generator such as a linear congruential generator (which uses a seed, multiplies by a constant, adds another constant, and takes the remainder after dividing by a modulus to produce a sequence).
• The Box-Muller Transformation maps the pair of uniform samples (x, y) to a pair of normally distributed values (z1, z2) using the two equations above.
• The transformation is based on the source statement that “the sum of two independent uniformly distributed random variables has a normal distribution” and leverages trigonometric functions with a radial transform to produce normally distributed outputs.
• The method is simple to implement and can be integrated into statistical and mathematical applications that require normally distributed random samples.

Examples

Example with x = 0.5 and y = 0.5

Using x = 0.5 and y = 0.5 and the equations above:

$$z_1 = \sqrt{-2\ln(0.5)}\cos(2\pi \cdot 0.5) = 0.73\cos(\pi) = -0.73$$ $$z_2 = \sqrt{-2\ln(0.5)}\sin(2\pi \cdot 0.5) = 0.73\sin(\pi) = 0$$

Thus the two normally distributed random numbers obtained are -0.73 and 0, with a mean of 0 and a standard deviation of 1 in this example.

Use cases

• Generating normally distributed random numbers for statistical and mathematical applications.
• Applicable when many real-world phenomena that are modeled follow a normal distribution.

Related terms

• Linear congruential generator
• Uniform distribution
• Normal distribution