Uniform Distribution (U₍a,b₎) ○◂ Entries

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Uniform Distribution (U₍a,b₎) ○◂｜Definition｜1st｜20260604231540-00-⌔

Continuous uniform distribution - Wikipedia

Continuous uniform distribution

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In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds.¹ The bounds are defined by the parameters, $a$ and $b,$ which are the minimum and maximum values. The interval can either be closed (i.e. $[a,b]$) or open (i.e. $(a,b)$).² Therefore, the distribution is often abbreviated $U(a,b),$ where $U$ stands for uniform distribution.¹ The difference between the bounds defines the interval length; all intervals of the same length on the distribution’s support are equally probable. It is the maximum entropy probability distribution for a random variable $X$ under no constraint other than that it is contained in the distribution’s support.³

Printed 2026-06-28.

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Footnotes

1. Dekking, Michel (2005). A modern introduction to probability and statistics: understanding why and how. London, UK: Springer. pp. 60–61. ISBN 978-1-85233-896-1. ↩ ↩²

2. Walpole, Ronald; Myers, Raymond H.; Myers, Sharon L.; Ye, Keying (2012). Probability & Statistics for Engineers and Scientists. Boston, USA: Prentice Hall. pp. 171–172. ISBN 978-0-321-62911-1. ↩

3. Park, Sung Y.; Bera, Anil K. (2009). “Maximum entropy autoregressive conditional heteroskedasticity model”. Journal of Econometrics. 150 (2): 219–230. CiteSeerX 10.1.1.511.9750. doi:10.1016/j.jeconom.2008.12.014. ↩

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