Discrete Uniform Distribution

Compact study note.

Summary

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

Prerequisites

• Discrete Random Variable

Definition

$X$ is discrete uniform on $\{x_1,\dots ,x_n\}$ when each support point has probability $1/n$ .

Notation and Assumptions

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

Parameters

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

Support

$\{x_1,\dots ,x_n\}$ .

PMF or PDF

$P(X=x_i)=1/n$ for $i=1,\dots ,n$ .

CDF

$F_X(x)=n^{-1}\mathrm{\#}\{i:x_i\le x\}$ .

Moments

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

Essential Result

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

Small Example

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

Common Mistakes

• Assuming all finite outcomes are equally likely without justification.
• Confusing discrete uniform with continuous uniform.

Connections

• Binomial Distribution
• Continuous Uniform Distribution
• Discrete Distributions

References