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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

LearnYMA01WST02 · Statistics 2Continuous distributionsThe continuous uniform distribution

Continuous distributions · Subtopic 1 of 2

The Continuous Uniform Distribution

If $X$ is equally likely to be anywhere between $a$ and $b$ , then $X\sim U[a,b]$ . Its pdf is a flat rectangle of height $\dfrac1{b-a}$ , so any probability is just (width of the interval) ÷ (total width). Its mean is the midpoint and its variance is $\dfrac{(b-a)^2}{12}$ .

Edexcel YMA01 WST02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 30 minRevision 6 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Write down the pdf of $U[a,b]$ .
Find probabilities as areas of rectangles.
Use $E(X)=\tfrac{a+b}2$ and $\text{Var}(X)=\tfrac{(b-a)^2}{12}$ .
Model situations such as random cut points and rounding errors.

Pause and think

A bus comes every $20$ minutes and you arrive at a random time. What's the chance you wait less than $5$ minutes? Exactly $5$ minutes?

Section 2 of 14

Core idea

Start hereArriving at a random time

Buses come every 10 minutes and you turn up at a random moment. Your wait is anywhere from 0 to 10 minutes, every value equally likely. Its pdf is flat: a rectangle. Probabilities become simple length ratios.

The big idea

$X\sim U[a,b]$ : $f(x)=\frac1{b-a}$ . $P(c<X<d)=\frac{d-c}{b-a}$ . $E(X)=\frac{a+b}2$ , $\operatorname{Var}(X)=\frac{(b-a)^2}{12}$ .

1. Probabilities

Waiting time U[0, 10]: P(X < k) is a rectangle

k = 1.0

P(X<k)=\dfrac{k}{10}

: the chance of waiting less than 3 minutes is

0.3

f(x)=\frac1{b-a}\ (a\le x\le b),\qquad P(c<X<d)=\frac{d-c}{b-a}

$a,b$: end points.
$\frac1{b-a}$: height making the area 1.

Worked exampleWaiting time $U[0,10]$

$P(X<3)$ , $P(4<X<7)$ , mean, variance.

Goal: Four values.
Strategy: Length ratios; formulas.

$P(X<3)=\tfrac3{10}=0.3$
Rectangle width 3.
$P(4<X<7)=\tfrac3{10}=0.3$
Same width, same probability.
$E=5,\ \operatorname{Var}=\tfrac{100}{12}=8.33$
Formulas.

Answer

0.3

0.3

5

8.33

✓ CheckMean at the centre by symmetry ✓.

2. Deriving the variance

\operatorname{Var}(X)=\frac{(b-a)^2}{12}

$E(X^2)=\int_a^b\frac{x^2}{b-a}dx=\frac{b^3-a^3}{3(b-a)}$
Integrate.
$=\frac{a^2+ab+b^2}3$
$b^3-a^3=(b-a)(a^2+ab+b^2)$ .
$\operatorname{Var}=\frac{a^2+ab+b^2}3-\frac{(a+b)^2}4$
Subtract the squared mean.
$=\frac{4a^2+4ab+4b^2-3a^2-6ab-3b^2}{12}=\frac{(b-a)^2}{12}$
Common denominator; $a^2-2ab+b^2$ .

Worked exampleBreaking a rod

A 1 m rod breaks at a random point. P(shorter piece < 0.2 m)?

Goal: Probability.
Strategy: Break point $X\sim U[0,1]$ ; identify the region.

$X<0.2\ \text{or}\ X>0.8$
Shorter piece is near either end.
$0.2+0.2=0.4$
Two lengths.

Answer

0.4

✓ CheckForgetting the second end gives

0.2

: half the answer ✓.

Edge caseEndpoints

P(X=3)=0

, so

<

and

\le

give the same probability.

Summary

Flat pdf; probability = length ratio.
Mean midpoint; variance $(b-a)^2/12$ .

Rectangular distribution: Another name for continuous uniform.

X ~ U[2, 6]

The density is a rectangle of height

\dfrac{1}{b-a}=\dfrac14

, so the total area is

1

. Probabilities are areas:

P(3<X<5)=2\times\dfrac14=0.5

HistoryBuffon's needle. In 1777 the Comte de Buffon dropped needles onto a floor of parallel boards. Assuming the needle's position and angle were uniformly random, he showed the chance of crossing a line involves

\pi

, so you can estimate

\pi

by throwing needles.

Fun factRandom numbers on a computer. Programming languages' basic random function gives a number from

U[0,1]

. Almost every other random simulation (dice, normal distributions, games) is built by transforming those uniform numbers.

Section 3 of 14

Key formulae

PDF

f(x)=\frac1{b-a},\quad a\le x\le b

Mean

E(X)=\frac{a+b}2

Variance

\text{Var}(X)=\frac{(b-a)^2}{12}

TrickWidth over width. Every uniform probability is "width of what you want ÷ width of everything". No integration needed: draw the rectangle and measure.

Section 4 of 14

Worked examples

Example 1 · A probability0 / 1 steps

X\sim U[2,10]

. Find

P(X<5)

Example 2 · Finding a and b0 / 2 steps

X\sim U[a,b]

with

E(X)=6

and

\text{Var}(X)=3

. Find

a

and

b

Section 5 of 14

Explore the graph

Probability as a rectangle

Loading the interactive graph…

Try this

The rectangle is the pdf of $U[a,b]$ ; the darker part is $P(X<c)$ . The value is $\tfrac{c-a}{b-a}$ .
Make the interval wider. What happens to the height, and why must it?
Put $c$ at the midpoint. What probability do you get?

Height $\tfrac1{b-a}$ makes the total area exactly $1$ .

In the real worldRounding errors. If a length is rounded to the nearest centimetre, the rounding error is modelled as

U[-0.5,0.5]

: any error in that range is equally likely. Engineers use its variance,

\tfrac1{12}

, to estimate how errors add up.

Section 6 of 14

Common mistakes

Incorrect

P(X<5)=\tfrac5{10}

for

U[2,10]

Correct

\tfrac{5-2}{10-2}

Measure from the lower limit.

Incorrect

\text{Var}=\dfrac{b-a}{12}

Correct

\dfrac{(b-a)^2}{12}

Square the width.

Incorrect

P(X=5)=\tfrac18

Correct

0

A single point has no width.

IncorrectWidth with negative limits

Correct

b-a

, e.g.

5-(-3)=8

Subtracting a negative adds.

Continuous uniform distribution

A rectangle with a probability of exactly $1$ inside it and absolutely no drama

Section 7 of 14

Exam tip

Exam tipDefine the variable. In modelling questions, start with "Let

X

be the distance of the cut from

A

;

X\sim U[0,30]

". Stating the model clearly is usually the first mark.

Section 8 of 14

Guided practice

SA-IAL-S2-CD-G012 marksL1 FoundationCalculatorUniform probabilities

X\sim U[2,10]

. Find

P(X<5)

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-S2-CD1-0011 markL1 FoundationNon-calculatorUniform probabilities

X\sim U[0,5]

. Write down the value of the probability density function for

0\le x\le5

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD-0032 marksL1 FoundationCalculatorUniform probabilities

X\sim U[0,20]

. Find

P(4<X\le9)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD1-0051 markL1 FoundationNon-calculatorUniform probabilities

X\sim U[0,10]

. Find

P(X>7.5)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD1-0021 markL1 FoundationNon-calculatorMean & variance

X\sim U[3,11]

. Write down

E(X)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD1-0031 markL1 FoundationNon-calculatorMean & variance

X\sim U[0,12]

. Find

\text{Var}(X)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD-0013 marksL1 FoundationCalculatorMean & variance

X\sim U[-3,5]

. Find

E(X)

and

\text{Var}(X)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD-0053 marksL2 StandardCalculatorMean & variance

X\sim U[1,7]

. Find

E(X^2)

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S2-CD-0025 marksL3 ExamCalculatorMean & variance

X\sim U[a,b]

, where

E(X)=6

and

\text{Var}(X)=3

. Find

a

and

b

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-IAL-S2-CD-0043 marksL3 ExamCalculatorUniform probabilities

A rod of length

30

cm is cut at a point chosen at random along its length. Find the probability that the shorter piece is less than

6

cm long.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$f(x)=\tfrac1{b-a}$ on $[a,b]$ .
Probability $=$ width ÷ total width; $P(X=c)=0$ .
$E(X)=\tfrac{a+b}2$ , $\text{Var}(X)=\tfrac{(b-a)^2}{12}$ .
$E(X^2)=\text{Var}(X)+[E(X)]^2$ .

Section 12 of 14

Quick quiz

Q1 · SA-IAL-S2-CD-Q011 markL1 FoundationCalculatorMean & variance

For

X\sim U[a,b]

\text{Var}(X)=

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-IAL-S2-CD1-0041 markL1 FoundationNon-calculatorUniform probabilities

X\sim U[2,8]

. Write down

P(X=5)

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WST02 (Statistics 2)
Where: S2 · 3 Continuous distributions: the continuous uniform (rectangular) distribution
Expected: The continuous uniform distribution U[a, b]: its pdf, probabilities as areas, mean (a + b)/2 and variance (b − a)²/12; modelling, e.g. rounding errors or a random cut point.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: "A rod is cut at a random point… find the probability that the shorter piece is less than 6 cm."
Open in: YMA01 · WST02 9FM0 · Option

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).