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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9709Paper 6 · Probability & Statistics 2Continuous random variablesProbability density functions

Continuous random variables · Subtopic 1 of 3

Probability Density Functions

For a continuous variable, probability lives in area. A probability density function is a curve whose total area is $1$ ; the probability of landing between $a$ and $b$ is the area under the curve between them, found by integration.

Cambridge 9709 Paper 6Cambridge 9231 Paper 4Edexcel YMA01 WST02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 35 minRevision 6 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Know the properties of a pdf: $f(x)\ge0$ and $\int f(x)\,dx=1$ .
Find an unknown constant $k$ in a pdf.
Find probabilities as $P(a<X<b)=\int_a^b f(x)\,dx$ .
Sketch a pdf and find its mode.

Pause and think

A bus arrives at a random moment in a $10$ -minute window. What is the probability it arrives at exactly $3$ minutes $0.000\ldots$ seconds? What about between $3$ and $4$ minutes?

Section 2 of 14

Core idea

Start hereThe probability of exactly 1.5 seconds

A runner's reaction time could be $0.151$ s, $0.1512$ s, $0.15117$ s... infinitely many values. The chance of hitting any one exactly is zero. So for continuous quantities, probability lives in intervals, measured as area under a curve.

The big idea

A probability density function $f(x)\ge0$ has total area $1$ . $P(a<X<b)=\int_a^bf(x)\,dx$ . $P(X=a)=0$ , so $<$ and $\le$ are interchangeable.

1. Area is probability

f(x) = 3x²/8 for 0 ≤ x ≤ 2: area up to b

b = 0.30

P(X<b)=\displaystyle\int_0^b\frac{3x^2}{8}\,dx=\frac{b^3}{8}

. At

b=2

the whole area is

1

P(a<X<b)=\int_a^bf(x)\,dx,\qquad\int_{-\infty}^{\infty}f(x)\,dx=1

$f(x)$: density: probability per unit of $x$ (can exceed 1).

Worked example $f(x)=kx^2$ , $0\le x\le3$

Find $k$ and $P(X>2)$ .

Goal: $k$ ; probability.
Strategy: Total area $1$ .

$k\left[\tfrac{x^3}3\right]_0^3=9k=1\Rightarrow k=\tfrac19$
Normalise.
$P(X>2)=\tfrac19\left[\tfrac{x^3}3\right]_2^3=\tfrac{27-8}{27}=\tfrac{19}{27}$
Integrate.

Answer

k=\frac19

;

\frac{19}{27}

✓ CheckMost area is near

3

(density grows), so

>\frac13

of the range having

70\%

is plausible ✓.

Edge caseDensity above 1

f(x)=2

[0,0.5]

is valid: area

2\times0.5=1

Common misconceptionReading f as a probability

f(1)

is not

P(X=1)

(that's

0

Pause and think

$f(x)=\frac38x^2$ on $[0,2]$ . $P(X<1)$ ?

Summary

Probability = area.
Total area 1 fixes constants.

pdf: Probability density function.

A probability density function: f(x) = ¾x(2 − x) for 0 ≤ x ≤ 2

A pdf is never negative and the total area under it is

1

. Probabilities are areas:

P(a<X<b)=\int_a^b f(x)\,dx

HistoryLaplace and the density. Pierre-Simon Laplace's Théorie analytique des probabilités (1812) treated probability with calculus, using smooth density curves and integrals instead of counting. Laplace called probability theory "common sense reduced to calculation".

Fun factProbability zero is not impossible. Your exact height to infinitely many decimal places has probability

0

of being any particular number, yet you have a height. For continuous variables "probability

0

" and "impossible" are different things, a subtlety that troubled mathematicians until Andrey Kolmogorov put probability on firm foundations in 1933.

Section 3 of 14

Key formulae

Properties

f(x)\ge0,\qquad\int_{-\infty}^{\infty}f(x)\,dx=1

Probabilities

P(a<X<b)=\int_a^bf(x)\,dx

Mode

f'(x)=0\ \text{(or an end point if } f \text{ is monotonic)}

TrickUse shapes before integrals. If the pdf is a straight line or constant, the area is a triangle, rectangle or trapezium: no integration needed. For

f(x)=\tfrac x8

[0,4]

P(X<2)

is a triangle of base

2

and height

\tfrac14

: area

\tfrac14

. And if the pdf is symmetric, half the area lies on each side of the centre.

Section 4 of 14

Worked examples

Example 1 · Finding k0 / 2 steps

f(x)=k(x+1)

for

0\le x\le2

, and

0

otherwise. Find

k

Example 2 · A probability0 / 2 steps

f(x)=\tfrac38x^2

for

0\le x\le2

. Find

P(X>1)

Example 3 · The mode0 / 2 steps

f(x)=\tfrac3{32}x(4-x)

for

0\le x\le4

. Find the mode.

Section 5 of 14

Explore the graph

Area is probability

Loading the interactive graph…

Try this

The curve is $f(x)=kx^2$ on $[0,3]$ ; the shaded area is $P(X<a)=\tfrac{k a^3}{3}$ .
Only one value of $k$ makes the total area $1$ . Set $a=3$ : the area is $9k$ , so $k=\tfrac19\approx0.11$ .
With $k\approx\tfrac19$ and $a=2$ , the shaded area is $\tfrac{8}{27}\approx0.30$ . Does that match the picture?

A pdf must enclose exactly one unit of area.

In the real worldWaiting for the bus. Transport planners model waiting times with pdfs. For buses every

10

minutes arriving at random times, the waiting time is uniform on

[0,10]

f(x)=0.1

. If buses bunch up, the pdf changes shape and the average wait gets longer, even with the same number of buses per hour.

Section 6 of 14

Common mistakes

Incorrect

P(X=2)=f(2)

Correct

P(X=2)=0

f

is a density; probabilities need an interval.

Incorrect"

f(x)>1

, so it can't be a pdf"

CorrectDensities can exceed

1

Only the total area must be

1

IncorrectIntegrating outside the stated interval

CorrectUse only where

f(x)\ne0

f(x)=0

elsewhere.

IncorrectMode

=

value where

f(x)=0

CorrectWhere

f(x)

is greatest

The mode is the peak of the curve.

$f(2)=1.5$ . Me: "a probability of $1.5$ ?!"

The pdf: "I'm a density, not a probability. Relax."

Section 7 of 14

Exam tip

Exam tipLabel sketches. A pdf sketch needs the shape, the end points of the interval labelled on the

x

-axis, and the key

f

values (for example the maximum). Draw

f(x)=0

outside the interval too, or say so.

Section 8 of 14

Guided practice

SA-9709-P6-CR-G012 marksL1 FoundationCalculatorPDFs

X

has probability density function

f(x)=kx^2

for

0\le x\le3

, and

0

otherwise. Find

k

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P6-CR1-0012 marksL1 FoundationCalculatorPDFs

f(x)=kx

for

0\le x\le4

, and

0

otherwise, is a probability density function. Find

k

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR-0012 marksL1 FoundationCalculatorPDFs

For

f(x)=\tfrac{x^2}9

0\le x\le3

, find

P(X<2)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR1-0022 marksL1 FoundationCalculatorPDFs

f(x)=\tfrac38x^2

for

0\le x\le2

, and

0

otherwise. Find

P(X>1)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR-0044 marksL2 StandardCalculatorPDFs

f(x)=k(4-x)

for

0\le x\le4

, and

0

otherwise. Find

k

and

P(X>1)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR1-0053 marksL2 StandardCalculatorPDFs

f(x)=k(x+1)

for

0\le x\le2

, and

0

otherwise. Find

k

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR1-0031 markL1 FoundationCalculatorSketching pdfs

f(x)=0.25

for

1\le x\le5

, and

0

otherwise. Find

P(2<X<3.5)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR-0073 marksL2 StandardCalculatorPDFs

f(x)=kx(4-x)

for

0\le x\le4

, and

0

otherwise. Find

k

, and state

E(X)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-CR1-0062 marksL2 StandardCalculatorSketching pdfs

f(x)=\tfrac3{32}x(4-x)

for

0\le x\le4

, and

0

otherwise. Find the mode of

X

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-9709-P6-CR-0083 marksL3 ExamCalculatorPDFs

For

f(x)=\tfrac3{32}x(4-x)

0\le x\le4

, find

P(1<X<3)

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$f(x)\ge0$ and total area $1$ .
$P(a<X<b)=\int_a^bf(x)\,dx$ ; $P(X=a)=0$ .
Find $k$ by setting the total integral equal to $1$ .
Mode: where $f$ is largest.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P6-CR-Q011 markL1 FoundationCalculatorPDFs

Which is a property of every probability density function?

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P6-CR-0111 markL1 FoundationCalculatorSketching pdfs

For a continuous random variable

X

P(X=2)

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-9709-P6-CR1-0041 markL1 FoundationCalculatorSketching pdfs

Which of these cannot be a probability density function on the given interval?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 9709 AS & A Level Mathematics
Where: Paper 6, Probability & Statistics 2 · 6.3 Continuous random variables
Expected: Understand the concept of a continuous random variable, and recall and use properties of a probability density function (restricted to functions defined over a single interval); use a pdf to find probabilities.
Calculator: Permitted. Give non-exact answers to 3 significant figures; probabilities may be fractions or decimals.
Typical style: f(x) = kx(4 − x) on an interval: find k, sketch the pdf, then find a probability by integration.
Open in: 9709 · Paper 6 9231 · Paper 4

Section 14 of 14

Past paper questions

5 questions (47 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/62 · Oct/Nov 2025Question 59 marks
Find a constant in a probability density function, then use the median for a probability.
Question paper, page 6 Mark scheme
9709/62 · May/June 2025Question 79 marks
Probability density function: find a constant, a parameter from the mean, and the median.
Question paper, page 10 Mark scheme
9709/62 · Oct/Nov 2024Question 611 marks
Continuous distribution: constants and the median.
Question paper, page 8 Mark scheme
9709/62 · May/June 2024Question 710 marks
Probability density function: a constant, the median and E(X).
Question paper, page 10 Mark scheme
9709/62 · May/June 2023Question 78 marks
Linear density function: constant, expectation and a limit from the median.
Question paper, page 12 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9709, 2026–2027 (official specification).