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The cumulative distribution function

Edexcel IAL Mathematics · WST02 Statistics 2

The cumulative distribution function F(x) gives P(X ≤ x). Once you have it, probabilities, medians and quartiles come straight from it.

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Definition

F(x) = P(X ≤ x) = ∫ from the lowest value to x of f(t) dt.

Use a different letter (t) inside the integral, because x is the upper limit. F(x) rises from 0 to 1 and never decreases. P(a < X ≤ b) = F(b) - F(a).

Writing F in full

Examiners expect the cdf for every real x, including the lines where it is 0 and 1. For f(x) = 3x2/8 on 0 ≤ x ≤ 2:

F(x) = 0 for x < 0

F(x) = x3/8 for 0 ≤ x ≤ 2

F(x) = 1 for x > 2

Leaving out the 0 or 1 line costs a mark.

Piecewise pdfs

For a pdf in two pieces, integrate the first piece as normal. For the second piece, add the total probability of the first piece to the integral from the joining point: F(x) = F(2) + ∫ from 2 to x of f(t) dt. Check that F reaches 1 at the top of the range and that the pieces agree at the join.

Rangef(x)
0 ≤ x ≤ 2x/4
2 < x ≤ 4(4 - x)/4
otherwise0

Worked example

Find the cdf of the pdf in the table.

  1. For 0 ≤ x ≤ 2: F(x) = ∫ from 0 to x of t/4 dt = x2/8, so F(2) = 1/2.
  2. For 2 < x ≤ 4: F(x) = 1/2 + ∫ from 2 to x of (4 - t)/4 dt = 1/2 + x - x2/8 - 3/2 = x - x2/8 - 1.
  3. Check: F(4) = 4 - 2 - 1 = 1.

Answer: F(x) = 0 for x < 0; x2/8 for 0 ≤ x ≤ 2; x - x2/8 - 1 for 2 < x ≤ 4; 1 for x > 4

Key idea: F(x) = P(X ≤ x) = ∫ f(t) dt up to x. Write F for all x, with the 0 and 1 lines. Piecewise: carry the earlier total forward.

Check yourself

  1. For the pdf in the table, find F(1). Give your answer as a fraction in its simplest form.
    Show answer

    1/8. F(1) = 12/8 = 1/8.

  2. For the pdf in the table, find F(3). Give your answer as a fraction in its simplest form.
    Show answer

    7/8. F(3) = 3 - 9/8 - 1 = 7/8.

  3. For the pdf in the table, find P(1 < X ≤ 3). Give your answer as a fraction in its simplest form.
    Show answer

    3/4. F(3) - F(1) = 7/8 - 1/8 = 3/4.

  4. f(x) = 3x2/8 for 0 ≤ x ≤ 2 and 0 otherwise. Using F(x) = x3/8, find P(X ≤ 1.5). Give your answer as a fraction in its simplest form.
    Show answer

    27/64. F(1.5) = 3.375/8 = 27/64.

  5. f(x) = 2x/9 for 0 ≤ x ≤ 3 and 0 otherwise. Find F(2). Give your answer as a fraction in its simplest form.
    Show answer

    4/9. F(x) = ∫ from 0 to x of 2t/9 dt = x2/9, so F(2) = 4/9.

  6. A student writes the cdf of f(x) = 3x2/8 (0 ≤ x ≤ 2) as 'F(x) = x3/8 for 0 ≤ x ≤ 2'. Why would this lose a mark?
    • It does not give F(x) = 0 for x < 0 and F(x) = 1 for x > 2
    • x3/8 is the wrong expression
    • F should be written in terms of t
    • The range should be 0 < x < 8
    Show answer

    It does not give F(x) = 0 for x < 0 and F(x) = 1 for x > 2. The cdf must be defined for all real x; examiners report students losing the mark for missing these lines.

  7. Which statement is true for every cumulative distribution function F(x)?
    • F(x) can be negative
    • F(x) is the gradient of f(x)
    • F(x) never decreases as x increases
    • F(x) always equals 1 at x = 0
    Show answer

    F(x) never decreases as x increases. F(x) adds up probability as x increases, so it cannot go down.

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