Uniform

The cumulative distribution function $\func$UniformDist$\left(\vphantom{ x;a,b}\right.$x;a, b$\left.\vphantom{ x;a,b}\right)$ for a < b is the function

$\displaystyle \func$UniformDist$\displaystyle \left(\vphantom{ x;a,b}\right.$x;a, b$\displaystyle \left.\vphantom{ x;a,b}\right)$ = $\displaystyle \left\{\vphantom{
\begin{array}{ccc}
0 & if & x\leq a \\
\frac{x-a}{b-a} & if & a\leq x\leq b \\
1 & if & b\leq x
\end{array}
}\right.$$\displaystyle \begin{array}{ccc}
0 & if & x\leq a \\
\frac{x-a}{b-a} & if & a\leq x\leq b \\
1 & if & b\leq x
\end{array}$

The probability density function of the uniform distribution on an interval [a, b], where a < b, is the function

$\displaystyle \limfunc$UniformDen(x;a, b) = $\displaystyle \left\{\vphantom{
\begin{array}{ccc}
0 & if & x\leq a \\
\frac{1}{b-a} & if & a\leq x\leq b \\
0 & if & b\leq x
\end{array}
}\right.$$\displaystyle \begin{array}{ccc}
0 & if & x\leq a \\
\frac{1}{b-a} & if & a\leq x\leq b \\
0 & if & b\leq x
\end{array}$

The uniform random variable is the continuous version of ``choosing a number at random.'' The probability that a uniform random variable on $\left[\vphantom{
a,b}\right.$a, b$\left.\vphantom{
a,b}\right]$ will have a value in either of two subintervals of $\left[\vphantom{
a,b}\right.$a, b$\left.\vphantom{
a,b}\right]$ of equal length is the same.

The following plots show cumulative distribution functions $\func$UniformDist$\left(\vphantom{ x;a,b}\right.$x;a, b$\left.\vphantom{ x;a,b}\right)$ and probability density functions $\limfunc$UniformDen(x;a, b) for (a, b) = $\left(\vphantom{ 0,1}\right.$0, 1$\left.\vphantom{ 0,1}\right)$, $\left(\vphantom{ 1.5,5}\right.$1.5, 5$\left.\vphantom{ 1.5,5}\right)$, $\left(\vphantom{ 3,15}\right.$3, 15$\left.\vphantom{ 3,15}\right)$ and -5≤x≤20.

itbpFU2.1577in1.7573in0inUniform density functions itbpFU2.1577in1.7573in0inUniform distribution functions