Discrete Uniform Distribution

Compact study note.

Summary

One discrete uniform distribution assigns equal probability to each value in a finite set.^[1]

$X$ is discrete uniform on ${x_{1}, \dots, x_{n}}$ when each support point has probability $1 / n$ .

The support is finite and all listed outcomes are equally likely.

$n \in {1, 2, \dots}$ .

${x_{1}, \dots, x_{n}}$ .

$P (X = x_{i}) = 1 / n$ for $i = 1, \dots, n$ .

$F_{X} (x) = n^{- 1} # {i : x_{i} \leq x}$ .

For support ${1, \dots, n}$ , $E [X] = (n + 1) / 2$ and $Var (X) = (n^{2} - 1) / 12$ .

Use this model only when equal likelihood is part of the experiment design or assumption.

One fair six-sided die has $P (X = k) = 1 / 6$ for $k = 1, \dots, 6$ .

OpenStax, Introductory Statistics 2e, "Chapter 4: Discrete Random Variables", https://openstax.org/books/introductory-statistics-2e/pages/4-introduction ↩︎