The discrete uniform distribution is a discrete probability distribution with probability function given by:

subject to the following conditions:

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

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

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

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

$\left\{\begin{array}{cc}0& u<a\\ \frac{1}{⌊b⌋-⌈a⌉+1}& u\le b\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ProbabilityFunction}\left(X\&comma;4\right)$

$\left\{\begin{array}{cc}0& 4<a\\ \frac{1}{⌊b⌋-⌈a⌉+1}& 4\le b\\ 0& \mathrm{otherwise}\end{array}\right.$

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

$\frac{⌈a⌉}{2}+\frac{⌊b⌋}{2}$

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

$\frac{{⌈a⌉}^2}{12}+\frac{{⌊b⌋}^2}{12}+\frac{⌊b⌋}{6}-\frac{⌈a⌉⌊b⌋}{6}-\frac{⌈a⌉}{6}$

$\mathrm{Variance}\left(X\right)\mathbf{assuming}a::\mathrm{integer},b::\mathrm{integer}$

$\frac{1}{12}{a}^2+\frac{1}{12}{b}^2+\frac{1}{6}b-\frac{1}{6}ab-\frac{1}{6}a$

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.