# 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, ${\displaystyle a}$ and ${\displaystyle b,}$ which are the minimum and maximum values. The interval can either be closed (i.e. ${\displaystyle [a,b]}$) or open (i.e. ${\displaystyle (a,b)}$).[2] Therefore, the distribution is often abbreviated ${\displaystyle U(a,b),}$ where ${\displaystyle U}$ 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 ${\displaystyle X}$ under no constraint other than that it is contained in the distribution's support.[3]
Notation Probability density functionUsing maximum convention Cumulative distribution function ${\displaystyle {\mathcal {U}}_{[a,b]}}$ ${\displaystyle -\infty ${\displaystyle [a,b]}$ ${\displaystyle {\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}}$ ${\displaystyle {\begin{cases}0&{\text{for }}xb\end{cases}}}$ ${\displaystyle {\tfrac {1}{2}}(a+b)}$ ${\displaystyle {\tfrac {1}{2}}(a+b)}$ ${\displaystyle {\text{any value in }}(a,b)}$ ${\displaystyle {\tfrac {1}{12}}(b-a)^{2}}$ ${\displaystyle {\tfrac {1}{4}}(b-a)}$ ${\displaystyle 0}$ ${\displaystyle -{\tfrac {6}{5}}}$ ${\displaystyle \ln(b-a)}$ ${\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{tb}-\mathrm {e} ^{ta}}{t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}$ ${\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{\mathrm {i} tb}-\mathrm {e} ^{\mathrm {i} ta}}{\mathrm {i} t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}$