Definition:Uniform Distribution/Continuous

Definition

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 known as

The continuous uniform distribution is also known as the rectangular distribution.

Also see

• Definition:Discrete Uniform Distribution

• Results about the continuous uniform distribution can be found here.

Linguistic Note

The name rectangular distribution for the continuous uniform distribution arises from the shape of its graph: it has the shape of a rectangle of height $\dfrac 1 {b - a}$.

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): uniform distribution: 2. (rectangular distribution)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): uniform distribution: 2. (rectangular distribution)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): uniform distribution
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions