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.

Probability density function

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

Probability density function

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.

Continuous uniform distribution