﻿ Probability distributions > Continuous univariate distributions > Uniform distribution

Uniform distribution

The Uniform distribution is perhaps the simplest of all continuous distributions since all values in the range [0,a] (say) are equally likely. The probability density function is simply

from which we can obtain the first two moments and hence the mean and variance extremely simply by integration:

The form of the distribution is a rectangle, and although the range could be over any interval it is customary to define it over the interval [0,a], as illustrated below:

Uniform distribution over the interval [0,a]

The distribution described above is continuous, but a discrete version also exists. For example, when all outcomes of a finite set are equally likely, as in the rolling of an unbiased die to give the values 1,2,3,4,5 or 6, each with probability 1/6. Key measures for the continuous Uniform distribution are provided in the table below and is noted above, can be obtained simply by integration:

Mean

a/2 or (b-a)/2

Variance

a2/12 or (b-a)2/12

Skewness

0

Kurtosis

-6/5

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, II. Houghton Mifflin/J Wiley & Sons, New York

Mathworld: Weisstein E W: Uniform distribution: https://mathworld.wolfram.com/UniformDistribution.html

Wikipedia: Uniform distribution: https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29