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Discrete Uniform Distribution — Definition, Formula & Examples

A discrete uniform distribution is a probability distribution where a finite number of distinct outcomes each have the same probability of occurring. Rolling a fair die is a classic example — each face has a probability of 1/6.

A discrete random variable XX follows a discrete uniform distribution on integers a,a+1,…,ba, a+1, \ldots, b if P(X=k)=1b−a+1P(X = k) = \frac{1}{b - a + 1} for every integer kk with a≤k≤ba \le k \le b, and P(X=k)=0P(X = k) = 0 otherwise.

Key Formula

P(X=k)=1n,μ=a+b2,σ2=n2−112P(X = k) = \frac{1}{n}, \qquad \mu = \frac{a + b}{2}, \qquad \sigma^2 = \frac{n^2 - 1}{12}
Where:
  • nn = Total number of equally likely outcomes, equal to $b - a + 1$
  • aa = Smallest possible value of $X$
  • bb = Largest possible value of $X$
  • μ\mu = Mean (expected value) of the distribution
  • σ2\sigma^2 = Variance of the distribution

Worked Example

Problem: A fair die is rolled once. Find the mean and variance of the outcome.
Identify parameters: The outcomes are 1 through 6, so a=1a = 1, b=6b = 6, and n=6n = 6.
Compute the mean: Use the mean formula for the discrete uniform distribution.
μ=a+b2=1+62=3.5\mu = \frac{a + b}{2} = \frac{1 + 6}{2} = 3.5
Compute the variance: Use the variance formula with n=6n = 6.
σ2=n2−112=36−112=3512≈2.917\sigma^2 = \frac{n^2 - 1}{12} = \frac{36 - 1}{12} = \frac{35}{12} \approx 2.917
Answer: The mean is 3.53.5 and the variance is 3512≈2.917\frac{35}{12} \approx 2.917.

Visualization

Why It Matters

The discrete uniform distribution serves as a baseline model whenever outcomes are equally likely, from lottery drawings to random number generators. It also appears as a building block in simulation and Monte Carlo methods, where uniformly distributed integers are transformed into samples from more complex distributions.

Common Mistakes

Mistake: Confusing the discrete uniform distribution with the continuous uniform distribution.
Correction: The discrete version assigns probability to individual integers, while the continuous version assigns probability density over a real-valued interval. Their variance formulas differ: (n2−1)/12(n^2 - 1)/12 vs. (b−a)2/12(b - a)^2/12.