Bayesian Statistics: An Introduction, 4th Edition

A.15 Uniform distribution

X has a uniform distribution on the interval (a, b) denoted

if it has density

where

is the indicator function of the set (a, b). The mean and variance are

There is no unique mode, but the distribution is symmetrical, and hence

Sometimes we have occasion to refer to a discrete version; Y has a discrete uniform distribution on the interval [a, b] denoted

if it has a discrete distribution with density

The mean and variance are

using formulae for the sum and sum of squares of the first n natural numbers [the variance is best found by noting that the variance of UD(a, b) equals that of UD(1, n) where n=b–a+1]. Again, there is no unique mode, but the distribution is ...