Discrete Uniform Distribution¶

Definition¶

Let $A=\{a_1,\ldots,a_n\}$ be a nonempty finite set of distinct real values. A variable is discrete uniform on $A$ if $p_X(a_i)=1/n$ for each $i$ and its PMF is zero outside $A$ .

The course's integer-range case has $A=\{a,a+1,\ldots,b\}$ $1, \dots, b}$ , where $a\le b$ are integers, and

p_X(k)=\frac{1}{b-a+1},\qquad k=a,\ldots,b.

The mean and variance formulas below apply to this consecutive-integer case; for an arbitrary finite set, average its listed values and squared deviations.

Interpretation¶

A discrete uniform variable picks one value from a finite set, with every value equally likely. It is the random-variable form of naive probability, as for a fair die or a random integer from a range.

Six equal probability bars of height 1/6 for the faces of a fair die, with the mean 3.5 marked.

The figure shows a fair die, $A=\{1,\ldots,6\}$ ${1, \dots, 6}$ : six bars of equal height $\tfrac{1}{6}$ . The dashed line at $3.5$ marks the mean from the Key Results; it lies at the centre because the bars are placed symmetrically, and it is not itself a possible value. The variance $35/12$ in the figure is the formula $(n^2-1)/12$ with $n=6$ values.

Only the number of values matters for the probabilities: each mass is $1/n$ , and the probability of a set of values is the number of points of $A$ in that set divided by $n$ . For the integer range $a,\ldots,b$ there are $n=b-a+1$ values, which is where the denominator in the PMF comes from; a common slip is to count $b-a$ values, but $\{3,\ldots,7\}$ has $5$ values, not $4$ .

A sum of uniform variables need not be uniform: the total of two fair dice takes the $11$ values $2,\ldots,12$ , but $7$ arises from six of the $36$ equally likely pairs and $2$ only from $(1,1)$ (see sum of independent random variables). The uniform distribution on an interval is the continuous counterpart, spreading probability evenly over an interval instead of over finitely many points.

Key Results¶

$\mathbb{E}X=(a+b)/2$ $b) /2$ .
$\operatorname{Var}(X)=((b-a+1)^2-1)/12$ $1) /12$ .

Examples¶

A fair die has mean $3.5$ and variance $35/12$ .
A uniform integer on $\{-2,-1,0,1,2\}$ has $\mathbb{P}(X\ge1)=2/5$ and mean $0$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3
Nicky van Foreest · Probability Distributions · Refresher

Aliases: Dunif, discrete uniform random variable, discrete uniform

Last updated: 27 October 2026 11:52