All subjects · Mathematics · WST01 (S1)

The discrete uniform distribution

When every value of a discrete random variable is equally likely, it has a discrete uniform distribution. Its mean and variance have neat formulae.

Definition

X has a discrete uniform distribution over 1, 2, ..., n if P(X = x) = 1/n for each x. Example: the score on a fair six-sided die (n = 6).

Mean and variance

For X uniform on 1, 2, ..., n:

E(X) = (n + 1)/2 and Var(X) = (n2 - 1)/12.

For a fair die: E(X) = 3.5, Var(X) = 35/12. These formulae are in the formula booklet, but only for values 1 to n.

Other ranges

If the values are not 1 to n, link them to a variable that is. Example: Y is uniform on 5, 6, ..., 14. Then Y = X + 4 with X uniform on 1 to 10, so E(Y) = 5.5 + 4 = 9.5 and Var(Y) = Var(X) = 99/12 = 8.25.

For values 2, 4, 6, ..., 20, use Y = 2X with X on 1 to 10: E(Y) = 11, Var(Y) = 4 × 8.25 = 33.

Uniform probabilities: P(X ≤ k) = k/n for X on 1 to n.

Worked example

X is the score on a fair 10-sided die numbered 1 to 10. Find E(X) and Var(X).

  1. n = 10
  2. E(X) = 11/2
  3. Var(X) = (100 - 1)/12

Answer: E(X) = 5.5, Var(X) = 8.25

Key idea: Uniform on 1 to n: E = (n + 1)/2, Var = (n2 - 1)/12; otherwise rewrite as aX + b.

Check yourself

  1. X is uniform on 1, 2, ..., 20. Find E(X).
    Show answer

    10.5. E(X) = (20 + 1)/2 = 10.5.

  2. X is uniform on 1, 2, ..., 10. Find Var(X).
    Show answer

    8.25. (100 - 1)/12 = 8.25.

  3. X is the score on a fair six-sided die. Find Var(X). Give your answer as a fraction in its simplest form.
    Show answer

    35/12. (36 - 1)/12 = 35/12.

  4. X is uniform on 1, 2, ..., n and E(X) = 12. Find n.
    Show answer

    23. (n + 1)/2 = 12, so n = 23.

  5. Y is uniform on 5, 6, ..., 14. Find E(Y).
    Show answer

    9.5. Y = X + 4 with X uniform on 1 to 10: E(Y) = 5.5 + 4 = 9.5.

  6. Y takes the values 2, 4, 6, ..., 20, each equally likely. Find Var(Y).
    Show answer

    33. Y = 2X with X on 1 to 10: Var(Y) = 4 × 8.25 = 33.

  7. X is uniform on 1, 2, ..., 25. Find P(X ≤ 8). Give your answer as a fraction in its simplest form.
    Show answer

    8/25. 8 equally likely values out of 25: 8/25.

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Sources

Last updated 7 November 2026. Spotted a mistake? Tell us.