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. 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. ${\displaystyle (a,b)}$). Therefore, the distribution is often abbreviated ${\displaystyle U(a,b),}$ where ${\displaystyle U}$ stands for uniform distribution. 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.

The probability density function of the continuous uniform distribution is $${\displaystyle f(x)={\begin{cases}{\dfrac {1}{b-a}}&{\text{for }}a\leq x\leq b,\\[8pt]0&{\text{for }}x<a\ {\text{ or }}\ x>b.\end{cases}}}$$

The values of ${\displaystyle f(x)}$ at the two boundaries ${\displaystyle a}$ and ${\displaystyle b}$ are usually unimportant, because they do not alter the value of ${\textstyle \int _{c}^{d}f(x)dx}$ over any interval ${\displaystyle [c,d],}$ nor of ${\textstyle \int _{a}^{b}xf(x)\,dx,}$ nor of any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be ${\displaystyle {\tfrac {1}{b-a}}.}$ The latter is appropriate in the context of estimation by the method of maximum likelihood. In the context of Fourier analysis, one may take the value of ${\displaystyle f(a)}$ or ${\displaystyle f(b)}$ to be ${\displaystyle {\tfrac {1}{2(b-a)}},}$ because then the inverse transform of many integral transforms of this uniform function will yield back the function itself, rather than a function which is equal "almost everywhere", i.e. except on a set of points with zero measure. Also, it is consistent with the sign function, which has no such ambiguity.

Any probability density function integrates to ${\displaystyle 1,}$ so the probability density function of the continuous uniform distribution is graphically portrayed as a rectangle where ⁠${\displaystyle b-a}$⁠ is the base length and ⁠${\displaystyle {\tfrac {1}{b-a}}}$⁠ is the height. As the base length increases, the height (the density at any particular value within the distribution boundaries) decreases.

In terms of mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2},}$ the probability density function of the continuous uniform distribution is $${\displaystyle f(x)={\begin{cases}{\dfrac {1}{2\sigma {\sqrt {3}}}}&{\text{for }}-\sigma {\sqrt {3}}\leq x-\mu \leq \sigma {\sqrt {3}},\\[2pt]0&{\text{otherwise}}.\end{cases}}}$$

The cumulative distribution function of the continuous uniform distribution is: $${\displaystyle F(x)={\begin{cases}0&{\text{for }}x<a,\\[8pt]{\frac {x-a}{b-a}}&{\text{for }}a\leq x\leq b,\\[8pt]1&{\text{for }}x>b.\end{cases}}}$$

Its inverse is: $${\displaystyle F^{-1}(p)=a+p(b-a)\quad {\text{ for }}0<p<1.}$$

In terms of mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2},}$ the cumulative distribution function of the continuous uniform distribution is: $${\displaystyle F(x)={\begin{cases}0&{\text{for }}x-\mu <-\sigma {\sqrt {3}},\\{\frac {1}{2}}\left({\frac {x-\mu }{\sigma {\sqrt {3}}}}+1\right)&{\text{for }}-\sigma {\sqrt {3}}\leq x-\mu <\sigma {\sqrt {3}},\\1&{\text{for }}x-\mu \geq \sigma {\sqrt {3}};\end{cases}}}$$