Definition:Uniform Distribution

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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:

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


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 $a, b \in \R$, $a < b$.


$X$ 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.