Continuous uniform distribution

The continuous uniform distribution gives every value in an interval [a, b] the same density.

f(x) = \frac{1}{b-a}, \;\; a \le x \le b; \qquad F(x) = \frac{x-a}{b-a}

pdf and cdf on the interval

$a, b$: lower and upper ends of the interval ( $a < b$ )
$x$: a value in the interval

\E(X) = \frac{a+b}{2}, \qquad \Var(X) = \frac{(b-a)^2}{12}

Mean and variance

$X$: $X \sim \text{Uniform}(a, b)$

When to use

When every value in a range is equally likely: a random arrival time within a known window, rounding error, random numbers used in simulation. Probabilities are proportional to interval length.

How to interpret

$P(c \le X \le d) = (d - c)/(b - a)$ for $c, d$ inside the interval.
The quantile is $Q(p) = a + p(b - a)$ .

Common mistakes

Dividing by 12 without squaring the width in the variance.
Using the continuous uniform for a die (that is the discrete uniform on 1–6).

Example

A bus arrives at a uniformly random time in the next 20 minutes. $P(X \le 12) = 12/20 = 0.6$ ; the 90th percentile is 18 minutes; mean 10 minutes, SD $20/\sqrt{12} \approx 5.77$ minutes.