uniform distribution

Uniform distribution

A real random variable $X$ has a uniform distribution over a range $[a,b]$, denoted $X\sim \mathcal{U}[a,b]$, if it has the PDF

$$f_X(x)=\{\begin{array}{ll}\frac{1}{b-a}& a\le x\le b\\ 0& \text{otherwise.}\end{array}$$

The CDF is then the linear ramp $F_X(x)=\frac{x-a}{b-a}$ for $a\le x\le b$ (and $0$ below $a$, $1$ above $b$): all values in $[a,b]$ are equally likely, so probability accumulates at a constant rate.

Note

If $b-a<1$, the PDF $f_X(x)$ is $>1$, which is legal as long as the total area under the PDF is $1$.
Scale $c\cdot (b-a)⟹$ probability for a fixed interval $w$ drops by $\frac{1}{c}$ to $\frac{w}{c(b-a)}$.