# Definition:Continuous Uniform Distribution

## Definition

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
**the continuous uniform distribution**can be found here.

## Technical Note

The $\LaTeX$ code for \(\ContinuousUniform {a} {b}\) is `\ContinuousUniform {a} {b}`

.

When the arguments are single characters, it is usual to omit the braces:

`\ContinuousUniform a b`