Uniform distribution

A uniform distribution assigns equal probability to every outcome over a defined range. In the continuous case on [a, b], the density is f(x) = 1/(b−a); in the discrete case with n outcomes, each has probability 1/n.

ByHoang TruongUpdated July 3, 2026

See it move

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A single die roll splits into six equally likely faces, each carrying probability 1/6 — the bar divides into six equal segments. The mean is 3.5 and the variance is 35/12. The continuous version on [a, b] has the same flat density, 1/(b − a), across the whole range.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

f(x) = \frac{1}{b - a}, \quad a \le x \le b

Variables

$f (x)$: probability density function (per unit of x)
$a$: lower bound of the support (units of x)
$b$: upper bound of the support (units of x)
$x$: random variable value (units of x)

Continuous uniform distribution on [a, b]; the flat density reflects equal probability for every value in the range.

LaTeX

E[X] = \frac{a + b}{2}

Variables

$E [X]$: expected value (mean) (units of x)
$a$: lower bound (units of x)
$b$: upper bound (units of x)

LaTeX

\text{Var}(X) = \frac{(b - a)^2}{12}

Variables

$V a r (X)$: variance ((units of x)²)
$a$: lower bound (units of x)
$b$: upper bound (units of x)

Check yourself

PracticeCORE

A random variable X follows a continuous uniform distribution on the interval [4, 10]. What are its mean and variance?

Select an answer to check your understanding.