Box–Muller method
Topic of Simulation, part of Probability
Box–Muller method
| Type | Topic |
| Taught in | Simulation |
| Part | Probability |
Two independent uniforms U,V on (0,1) give two independent standard normals, Z_1=\sqrt{-2\log U}\cos(2\pi V) and Z_2=\sqrt{-2\log U}\sin(2\pi V). The polar coordinates of a standard bivariate normal are an exponential squared radius and a uniform angle, which is why it works.
In Simulation
From Simulation
Normal. There is no elementary inverse of \Phi. If a standard normal draw Z is already available, then \mu+\sigma Z is \operatorname{N}(\mu,\sigma^2). Two independent standard normals can be made from two independent uniforms (U,V) by the Box–Muller map
Z_1=\sqrt{-2\log U}\,\cos(2\pi V), \qquad Z_2=\sqrt{-2\log U}\,\sin(2\pi V).
Used in
- Taught in Simulation
- Also used in Functions of random variables and limit theorems
- All 4 questions on Simulation
See also
Multivariate normal distribution · Linear congruential generator · Inverse-transform method · Acceptance–rejection method · Random numbers · Draws from the usual families · Cauchy distribution · Random seed