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The continuous uniform distribution

Edexcel IAL Mathematics · WST02 Statistics 2

The continuous uniform (rectangular) distribution gives every value in an interval the same density. The specification asks you to derive its mean, variance and cdf.

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Definition

X ~ U[a, b] has pdf f(x) = 1/(b - a) for a ≤ x ≤ b and 0 otherwise. The graph is a rectangle of width b - a and height 1/(b - a), so its area is 1. Probabilities are rectangle areas: P(c < X < d) = (d - c)/(b - a) for a ≤ c < d ≤ b.

Deriving the cdf and mean

F(x) = ∫ from a to x of 1/(b - a) dt = (x - a)/(b - a) for a ≤ x ≤ b, with F = 0 for x < a and F = 1 for x > b.

E(X) = ∫ x/(b - a) dx = [x2/(2(b - a))] from a to b = (b2 - a2)/(2(b - a)) = (a + b)/2, as expected from symmetry.

Deriving the variance

E(X2) = ∫ x2/(b - a) dx = (b3 - a3)/(3(b - a)) = (a2 + ab + b2)/3.

Var(X) = (a2 + ab + b2)/3 - (a + b)2/4 = (4a2 + 4ab + 4b2 - 3a2 - 6ab - 3b2)/12 = (a2 - 2ab + b2)/12 = (b - a)2/12.

X ~ U[a, b]Formula
f(x)1/(b - a)
F(x)(x - a)/(b - a)
E(X)(a + b)/2
Var(X)(b - a)2/12

Worked example

X ~ U[2, 10]. Find P(X > 7), E(X) and Var(X).

  1. f(x) = 1/8 on 2 ≤ x ≤ 10.
  2. P(X > 7) = (10 - 7)/8 = 3/8
  3. E(X) = 6; Var(X) = 82/12 = 16/3

Answer: 3/8, 6 and 16/3

Key idea: U[a, b]: f = 1/(b - a), F = (x - a)/(b - a), mean (a + b)/2, variance (b - a)2/12.

Check yourself

  1. X ~ U[2, 10]. Find P(X > 7). Give your answer as a fraction in its simplest form.
    Show answer

    3/8. (10 - 7)/(10 - 2) = 3/8.

  2. X ~ U[2, 10]. Find E(X).
    Show answer

    6. (2 + 10)/2 = 6.

  3. X ~ U[2, 10]. Find Var(X). Give your answer as a fraction in its simplest form.
    Show answer

    16/3. (10 - 2)2/12 = 64/12 = 16/3.

  4. X ~ U[2, 10]. Find P(3 < X < 5). Give your answer as a fraction in its simplest form.
    Show answer

    1/4. (5 - 3)/8 = 1/4.

  5. X ~ U[2, 10]. Find E(X2). Give your answer to 3 significant figures.
    Show answer

    41.3. E(X2) = Var(X) + [E(X)]2 = 16/3 + 36 = 124/3 = 41.333.

  6. X ~ U[a, b] has mean 10 and variance 12. Find b.
    Show answer

    16. (b - a)2/12 = 12 gives b - a = 12; (a + b)/2 = 10 gives a + b = 20. So b = 16 (and a = 4).

  7. X ~ U[0, 8]. What is P(X = 5)?
    • 1/8
    • 5/8
    • 3/8
    • 0
    Show answer

    0. X is continuous, so a single value has probability 0.

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Last updated 8 November 2026. Spotted a mistake? Tell us.