Uniform distribution

For $a<b$ , the uniform distribution on the interval $[a,b]$ , written $\operatorname{Unif}[a,b]$ , is the probability distribution with density

f(x)=\frac{1}{b-a}\mathbf 1_{[a,b]}(x)

with respect to Lebesgue measure.

Properties

If $X\sim\operatorname{Unif}[a,b]$ , then $\mathbb E[X]=(a+b)/2$ and $\operatorname{Var}(X)=(b-a)^2/12$ . The phrase “uniform distribution” can also refer to the uniform probability measure on a finite set; that discrete convention is stated when needed.

References

Geoffrey Grimmett and David Stirzaker, Probability and Random Processes, 3rd ed., Oxford University Press, 2001, §3.4.