Factsheet: Continuous uniform distribution
Where to use: The continuous uniform distribution is used when all continuous values \(x\) in the interval \(a\) to \(b\) are equally likely. The random variable \(X\) represents the outcome.
Notation: \(X \sim \textrm{Uniform}(a,b)\) or \(X \sim U(a,b)\).
Parameters: Two real numbers \(a,b\), where
\(a\) is the minimum value of an outcome,
\(b\) is the maximum value of an outcome.
| Quantity | Value | Notes |
|---|---|---|
| Mean | \(\mathbb{E}(X) = \dfrac{a+b}{2}\) | |
| Variance | \(\mathbb{V}(X) = \dfrac{(b-a)^2}{12}\) | |
| \(\mathbb{P}(X=x)=\begin{cases} \dfrac{1}{b-a} & \textsf{if } a \leq x \leq b \\0 & \textsf{otherwise}\end{cases}\) | ||
| CDF | \(\displaystyle\mathbb{P}(X\leq x)=\begin{cases} 0 & \textsf{if } x< a \\\dfrac{x-a}{b-a} & \textsf{if } a\leq x\leq b \\1 & \textsf{if } x>b \end{cases}\) |
Example: A machine from Cantor’s Confectionery is programmed to chop long candy bars into pieces, each with a length between 30 millimetres to 50 millimetres. Due to variations in the machine, each continuous value between this interval is equally likely. This can be expressed as \(X \sim U(30,50)\). It means 30 is the minimum value and 50 is the maximum value, where all continuous values of \(X\) for \(30 \leq x \leq 50\) are equally likely.
Further reading
This interactive element appears in Overview: Probability distributions.
Version history
v1.0: initial version created 08/25 by tdhc.
- v1.1: deprecated shinylive apps replaced 09/26 by tdhc.