All subjects · Mathematics · WST01 (S1)

The cumulative distribution function

The cumulative distribution function F(x) gives the probability that X is at most x. Exam questions often give F instead of p and expect you to move between them.

Definition

F(x0) = P(X ≤ x0) = Σ p(x) for all x ≤ x0.

F increases as x increases and reaches 1 at the largest value of X.

x1234
P(X = x)0.10.20.30.4
F(x)0.10.30.61

From F back to p

P(X = x) = F(x) - F(x - 1) for integer values. So P(X = 3) = F(3) - F(2) = 0.6 - 0.3 = 0.3.

Also P(X > x) = 1 - F(x), and P(a < X ≤ b) = F(b) - F(a).

F given as a formula

If F(x) is a formula, use F(largest value) = 1 to find an unknown constant. Example: F(x) = (x + k)2 / 25 for x = 1, 2, 3. F(3) = 1 gives (3 + k)2 = 25, so k = 2 (k > 0).

Examiners report that a common error is treating F as a probability function and making F values add to 1: do not do that.

Worked example

F(x) = x2/16 for x = 1, 2, 3, 4. Find P(X = 3).

  1. P(X = 3) = F(3) - F(2)
  2. = 9/16 - 4/16

Answer: 5/16

Key idea: F(x) = P(X ≤ x); P(X = x) = F(x) - F(x - 1); F(max) = 1, not ΣF = 1.

Check yourself

  1. F(x) = x2/16 for x = 1, 2, 3, 4. Find P(X = 3). Give your answer as a fraction in its simplest form.
    Show answer

    5/16. F(3) - F(2) = 9/16 - 4/16 = 5/16.

  2. F(x) = (x + k)2/25 for x = 1, 2, 3, where k is a positive constant. Find k.
    Show answer

    2. F(3) = 1: (3 + k)2 = 25, so 3 + k = 5 and k = 2.

  3. F(x) = (x + 2)2/25 for x = 1, 2, 3. Find P(X = 2). Give your answer as a fraction in its simplest form.
    Show answer

    7/25. F(2) - F(1) = 16/25 - 9/25 = 7/25.

  4. X takes 1, 2, 3, 4 with probabilities 0.1, 0.2, 0.3, 0.4. Find F(2).
    Show answer

    0.3. F(2) = P(X ≤ 2) = 0.1 + 0.2 = 0.3.

  5. F(3) = 0.6 for a discrete random variable X. Find P(X > 3).
    Show answer

    0.4. P(X > 3) = 1 - F(3) = 0.4.

  6. F(1) = 0.1, F(2) = 0.3, F(3) = 0.6, F(4) = 1. Find P(1 < X ≤ 3).
    Show answer

    0.5. F(3) - F(1) = 0.5.

  7. F(x) for x = 1, 2, 3 is given in a table with an unknown k. Which equation finds k?
    • F(1) + F(2) + F(3) = 1
    • F(1) = 0
    • F(2) = 0.5
    • F(3) = 1
    Show answer

    F(3) = 1. F at the largest value is the total probability, 1.

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