Overview
The Box-Muller Transform converts two independent uniform samples into two independent standard normal samples.
CoreSource checked
Box-Muller Transform
Generates normally distributed random values from uniform random values.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Randomness & Sampling
- Difficulty
- Beginner
- Family
- Random Variate Generation
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 1
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
random drawsample selectionnormal samplerandom variate generationrandom sourcedistribution shape
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(1)
Average
O(1)
Worst
O(1)
Space
O(1)
When to Use
- Use Box-Muller Transform when you need to transform uniform random samples.
- Use it for simulation noise with inputs like two uniform random values.
- Use it when randomization, representative samples, or controlled variation are needed.
- Use it when the Random Variate Generation tradeoff is acceptable for the problem size.
When Not to Use
- Avoid it when reproducibility, distribution quality, or sample independence cannot be verified.
- Avoid it when the required output is not two normally distributed values.
What It Does
- Transforms uniform random samples
- Produces Gaussian random values
Use Cases
- Simulation noise
- Statistical sampling
- Monte Carlo models
Inputs
- Two uniform random values
Outputs
- Two normally distributed values
Key Formulas (1)
Normal sample
z0 = √−2 · ln(u1) · cos(2·π·u2)
z0 = sqrt(-2 * ln(u1)) * cos(2*pi*u2)Transforms two uniform random values into a standard normal sample.
Pseudocode
- 1.Draw two independent uniform values
- 2.Convert one value into a radius
- 3.Convert the other into an angle
- 4.Use sine and cosine to produce two normal samples
Canonical Example
- Input
- items = [A, B, C, D], seed = 7
- Output
- sampled or randomized result
- Explanation
- Box-Muller Transform: The algorithm uses controlled randomness to choose or generate values.
Implementation Notes
- Keep the input contract explicit: Two uniform random values.
- Separate the random number source from the algorithm so tests can use fixed seeds.
- Separate the random source from the sampler so tests can use fixed seeds and distribution checks.
Common Pitfalls
- Implementations must avoid taking the logarithm of zero.
- Sampling from the wrong distribution can make the estimate look precise but biased.
- Using biased random choices or forgetting to store the seed for reproducible runs.
- Handle empty, invalid, and boundary inputs before trusting two normally distributed values.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow uniform points mapped into a bell-shaped distribution.
Step 1 of 4: Draw two independent uniform values
uniform-to-normalNotes
Box-Muller Transform operates in O(1) time and O(1) space within the Random Variate Generation family. Key governing relation: z0 = sqrt(-2 * ln(u1)) * cos(2*pi*u2). Often compared with Linear Congruential Generator and Monte Carlo Simulation.
Sources (2)
What each reference supports for this entry.
Box-Muller Transform
PaperOriginal reference for generating normal variates from uniform random variables.
Open sourceNon-Uniform Random Variate Generation
BookRandom variate generation reference for Box-Muller Transform's distribution-sampling mechanism and acceptance or transform rules.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Has variantsMarsaglia Polar Method
Randomness & Sampling · Core
Marsaglia's polar method is a rejection-based Box-Muller variant that avoids explicit trigonometric functions.
Compare with thisAlternativesZiggurat Algorithm
Randomness & Sampling · Core
The ziggurat method and Box-Muller transform both generate nonuniform variates, especially normal variates, with different rejection/transform tradeoffs.
Compare with thisRelationships
View full relationship pageTransforms intoInverse Transform Sampling
Box-Muller transforms uniform random values into Gaussian samples.
Depends onLinear Congruential Generator
Box-Muller needs a source of uniform random values.
Related Algorithms
Tags
RandomnessSamplingNormal Distribution