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From cdf to pdf: f(x) = F'(x)

Edexcel IAL Mathematics · WST02 Statistics 2

Differentiating the cdf gives the pdf, and integrating the pdf gives the cdf. Questions often start with F(x) and ask for f(x), probabilities or a constant.

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The relationship

f(x) = dF(x)/dx wherever F is differentiable. So given a cdf, differentiate each piece to get the pdf, and give f(x) = 0 outside the range.

Example: F(x) = x2(3 - x)/4 = (3x2 - x3)/4 for 0 ≤ x ≤ 2 gives f(x) = (6x - 3x2)/4 = 3x(2 - x)/4.

Probabilities straight from F

P(a < X < b) = F(b) - F(a); P(X > a) = 1 - F(a). No integration is needed once you have F. Examiners warn against substituting a value into the wrong piece of a piecewise F: check which range the value lies in.

Finding constants in F

Use F(top of range) = 1 (and F(bottom) = 0). For F(x) = k(x2 + 2x) on 0 ≤ x ≤ 3: F(3) = 15k = 1, so k = 1/15. Also check that F is continuous at any join between pieces.

Worked example

F(x) = x2(3 - x)/4 for 0 ≤ x ≤ 2. Find P(0.5 < X < 1.5).

  1. F(1.5) = 2.25 × 1.5/4 = 0.84375
  2. F(0.5) = 0.25 × 2.5/4 = 0.15625
  3. Subtract

Answer: 0.6875 = 11/16

Key idea: f = F'. Probabilities come from F(b) - F(a). Use F(top) = 1 to find constants.

Check yourself

  1. F(x) = x2(3 - x)/4 for 0 ≤ x ≤ 2. Find f(1). Give your answer as a fraction in its simplest form.
    Show answer

    3/4. f(x) = (6x - 3x2)/4, so f(1) = 3/4.

  2. F(x) = x2(3 - x)/4 for 0 ≤ x ≤ 2. Find P(0.5 < X < 1.5). Give your answer as a fraction in its simplest form.
    Show answer

    11/16. F(1.5) - F(0.5) = 27/32 - 5/32 = 11/16.

  3. F(x) = x2(3 - x)/4 for 0 ≤ x ≤ 2. Find P(X > 1.2). Give your answer to 3 significant figures.
    Show answer

    0.352. 1 - F(1.2) = 1 - 1.44 × 1.8/4 = 0.352.

  4. F(x) = k(x2 + 2x) for 0 ≤ x ≤ 3, F(x) = 0 for x < 0 and F(x) = 1 for x > 3. Find k. Give your answer as a fraction in its simplest form.
    Show answer

    1/15. F(3) = k(9 + 6) = 15k = 1.

  5. F(x) = (x2 + 2x)/15 for 0 ≤ x ≤ 3. Find f(2). Give your answer as a fraction in its simplest form.
    Show answer

    2/5. f(x) = (2x + 2)/15, so f(2) = 6/15 = 2/5.

  6. F(x) = x2(3 - x)/4 for 0 ≤ x ≤ 2. Show that the median is 1 by finding F(1).
    Show answer

    0.5. F(1) = 1 × 2/4 = 0.5, so the median is 1.

  7. How do you get the pdf from the cdf?
    • Integrate F(x)
    • Divide F(x) by x
    • Differentiate F(x)
    • Subtract F(x) from 1
    Show answer

    Differentiate F(x). f(x) = dF/dx.

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