Uniform Distribution
Probability Distributions
Equal Probability Everywhere — Maximum Ignorance
The uniform distribution assigns equal probability density across an interval. It is the maximum entropy distribution — the most honest choice when you know nothing except the bounds.
- Random number generation — all PRNGs start with Uniform(0,1)
- Bayesian priors — "I have no prior information" → Uniform
- Rounding errors — modeling quantization noise
- Simulation — the source distribution for Monte Carlo methods
The uniform is the mother of all distributions — via the inverse transform, it births every other distribution.
Core Concepts
The continuous uniform distribution assigns equal probability density across an interval . It is the simplest continuous distribution and serves as the foundational building block from which all other continuous distributions can be derived via the inverse transform method.
Derivation of Mean and Variance
CDF and Its Properties
Higher Moments and Moment Generating Function
The Inverse Transform Method
Relationship to Other Distributions
- The order statistics of i.i.d. variables follow a Beta distribution: the -th order statistic is .
- The sum of i.i.d. variables follows the Irwin-Hall distribution, which approaches by the CLT.
- .
Specific Applications
- Random number generation — The is the source distribution for all Monte Carlo simulations and pseudo-random generators.
- Order statistics and ranking — The distribution of sorted uniform samples gives Beta distributions, used in nonparametric statistics.
- Bayesian noninformative priors — serves as a prior for probabilities in Bayesian inference (equivalent to ).
- Quantization and rounding errors — Modeling the distribution of continuous variables after discrete rounding.