# Definition:Uniform Distribution

## Definition

### Discrete Uniform Distribution

Let $X$ be a discrete random variable on a probability space.

Then $X$ has a discrete uniform distribution with parameter $n$ if and only if:

$\Img X = \set {1, 2, \ldots, n}$
$\map \Pr {X = k} = \dfrac 1 n$

That is, there is a number of outcomes with an equal probability of occurrence.

This is written:

$X \sim \DiscreteUniform n$

### Continuous Uniform Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $a, b \in \R$ such that $a < b$.

$X$ is said to be uniformly distributed on the closed real interval $\closedint a b$ if and only if it has probability density function:

$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}$

This is written:

$X \sim \ContinuousUniform a b$

## Also see

• Results about uniform distributions can be found here.