Continuous uniform distribution

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.

${\displaystyle {\mathbf {\text{Continuous uniform distribution}}}}$${\displaystyle {\mathbf {{\text{with parameters }}{\mathcal {a}}{\text{ and }}{\mathcal {b}}}}}$

Probability density function Using maximum convention
Cumulative distribution function
Notation	${\displaystyle {\mathcal {U}}_{[a,b]}}$
Parameters	${\displaystyle -\infty <a<b<\infty }$
Support	${\displaystyle [a,b]}$
PDF	${\displaystyle {\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}}$
CDF	${\displaystyle {\begin{cases}0&{\text{for }}x<a\\{\frac {x-a}{b-a}}&{\text{for }}x\in [a,b]\\1&{\text{for }}x>b\end{cases}}}$
Mean	${\displaystyle {\tfrac {1}{2}}(a+b)}$
Median	${\displaystyle {\tfrac {1}{2}}(a+b)}$
Mode	${\displaystyle {\text{any value in }}(a,b)}$
Variance	${\displaystyle {\tfrac {1}{12}}(b-a)^{2}}$
MAD	${\displaystyle {\tfrac {1}{4}}(b-a)}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle -{\tfrac {6}{5}}}$
Entropy	${\displaystyle \ln(b-a)}$
MGF	${\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{tb}-\mathrm {e} ^{ta}}{t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}$
CF	${\displaystyle {\begin{cases}{\frac {\mathrm {e} ^{\mathrm {i} tb}-\mathrm {e} ^{\mathrm {i} ta}}{\mathrm {i} t(b-a)}}&{\text{for }}t\neq 0\\1&{\text{for }}t=0\end{cases}}}$