The uniform distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The uniform variate is related to the unit parameter Beta variate by Uniform(0,1) ~ Beta(1,1).

The uniform variate is related to the unit parameter Power variate by Uniform(0,1) ~ Power(1,1).

Note that the Uniform command is inert and should be used in combination with the RandomVariable command.

For the discrete uniform distribution, see DiscreteUniform.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Uniform}\left(a\,b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\&lcub;\begin{array}{cc}0& u<a\\ \frac{1}{b-a}& u<b\\ 0& \mathrm{otherwise}\end{array}$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\&lcub;\begin{array}{cc}0.& 0.5<a\\ \frac{1}{b-1.a}& 0.5<b\\ 0.& \mathrm{otherwise}\end{array}$

$\mathrm{Mean}\left(X\right)$

$\frac{1}{2}a\&plus;\frac{1}{2}b$

$\mathrm{Variance}\left(X\right)$

$\frac{1}{12}{\left(b-a\right)}^2$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.