Table of Contents
What Is a Uniform Distribution?
The continuous uniform distribution, also called the rectangular distribution, is a probability distribution where all values within a given interval [a, b] are equally likely. The probability density function (PDF) is constant between a and b and zero elsewhere. It is the simplest continuous distribution and serves as a baseline for comparison with other distributions.
Uniform distributions model situations where there is equal likelihood across a range: random number generation, rounding errors, arrival times within a fixed interval, and many random sampling procedures. The standard uniform distribution U(0,1) is fundamental to Monte Carlo simulation and pseudo-random number generation in computing.
Formulas
Properties
| Property | Formula |
|---|---|
| Mean | (a + b) / 2 |
| Median | (a + b) / 2 |
| Mode | Any value in [a, b] |
| Variance | (b - a)² / 12 |
| Skewness | 0 (symmetric) |
| Kurtosis | -6/5 (platykurtic) |
Frequently Asked Questions
What is the standard uniform distribution?
The standard uniform distribution U(0,1) has a=0 and b=1. It is the basis of random number generation in computing. Any uniform distribution U(a,b) can be generated from U(0,1) using the transformation X = a + (b-a)*U.
How is the uniform distribution used in Monte Carlo simulation?
Monte Carlo methods use uniform random numbers to simulate complex systems. Using inverse transform sampling, U(0,1) random numbers can be converted to any other distribution. This makes the uniform distribution the foundation of all computational random sampling.