Mathematics 9709 · syllabus 6.3
Cambridge A Level Mathematics 9709 — Continuous random variables topical past papers
Probability density functions: finding a constant, probabilities, the mean, the variance, the median and percentiles by integrating over the variable's range.
- Questions
- 102
- Years
- 2012–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 37
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What Continuous random variables covers
A continuous random variable can take any value in an interval, and its distribution is described by a probability density function f(x). In Paper 6 (section 6.3) the density function is defined over a single interval, which may be infinite, such as f(x) = k/x³ for x ≥ 2. Probabilities are areas under the curve, found by integration, and the total area must be 1, which is how an unknown constant is found.
You calculate the mean E(X) = ∫x f(x) dx and the variance Var(X) = ∫x² f(x) dx − (E(X))², and find the median m, or another percentile, by solving an equation that sets the area up to m equal to ½, or to the proportion required. Sketching the density function shows its mode and whether the mean lies above or below the median. You are not asked to work with the cumulative distribution function as such.
Paper 6 assumes the calculus of Paper 3, so the integrals can involve exponentials or fractional powers as well as polynomials. The ideas are those of discrete random variables from Paper 5 with sums replaced by integrals, so E(X) and Var(X) are defined in the same way. The normal distribution is the best-known continuous distribution, though its areas come from tables rather than from integration.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 6.3 Continuous random variables
What you should be able to do
- Use the properties of a probability density function to find an unknown constant
- Find probabilities by integrating the density function over an interval
- Calculate E(X) and Var(X) from a density function
- Find the median, quartiles or other percentiles by solving an integral equation
- Sketch a density function and identify its mode
Key ideas
- f(x) ≥ 0 for all x, and the integral of f(x) over the whole range is 1
- P(a < X < b) = ∫ₐᵇ f(x) dx, and P(X = a) = 0 for any single value a
- E(X) = ∫x f(x) dx and Var(X) = ∫x² f(x) dx − (E(X))², over the whole range
- The median m satisfies ∫ f(x) dx = ½, integrating from the lower end of the range to m
How Continuous random variables has been examined
From 2022 to 2026, Continuous random variables was on Paper 6 · Probability & Statistics 2 in 14 of 14 exam series, worth about 18% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 6 · Probability & Statistics 2 | 37 | 18% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 8.
Filed in Momentum under: Continuous Random Variables.
A worked example
The random variable X has probability density function f(x) = kx² for 0 ≤ x ≤ 3, and f(x) = 0 otherwise. Find k, E(X), Var(X) and the median of X.
- The total probability is 1: ∫₀³ kx² dx = k[x³/3]₀³ = 9k = 1, so k = 1/9.
- E(X) = ∫₀³ x × x²/9 dx = [x⁴/36]₀³ = 81/36 = 2.25.
- E(X²) = ∫₀³ x⁴/9 dx = [x⁵/45]₀³ = 243/45 = 5.4, so Var(X) = 5.4 − 2.25² = 5.4 − 5.0625 = 0.3375.
- For the median m: ∫₀ᵐ x²/9 dx = m³/27 = ½, so m³ = 13.5.
- m = ∛13.5 = 2.381, which lies inside 0 ≤ x ≤ 3.
Answer: k = 1/9; E(X) = 2.25; Var(X) = 0.3375 (0.338 to 3 s.f.); median 2.38
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Forgetting to subtract (E(X))² when finding the variance.
- Integrating over the wrong limits: only the interval where the density is non-zero contributes, so the limits come from the stated range.
- Giving a median outside the range of X after solving a quadratic or cubic: check which root lies in the interval.
- Treating f(a) as the probability that X = a: for a continuous variable that probability is zero, and f(a) is a density.
Exam technique
- Show each integral with its limits, the integrated expression and the substitution of the limits; 'show that k = …' answers need every step.
- Sketch the density function first; it shows the range, the mode and roughly where the median should be.
- Give exact answers where the algebra allows, and otherwise 3 significant figures.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.