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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

Learn9FM0Option · Further Statistics 2Combinations of random variablesE and Var of aX + b and aX + bY

Combinations of random variables · Subtopic 1 of 3

Expectation and Variance of Combinations

Expectations behave exactly as you would hope: $E(aX+bY)=aE(X)+bE(Y)$ . Variances need care: constants are squared, added constants vanish, and for independent variables the variances always add, even for a difference.

Cambridge 9709 Paper 6Edexcel YMA01 WST03Edexcel 9FM0 Option≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Use $E(aX+b)=aE(X)+b$ and $\text{Var}(aX+b)=a^2\text{Var}(X)$ .
Use $E(aX+bY)=aE(X)+bE(Y)$ .
Use $\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)$ for independent $X$ , $Y$ .
Explain why $\text{Var}(X-Y)=\text{Var}(X)+\text{Var}(Y)$ .

Pause and think

$X$ is the time to walk to the bus stop and $Y$ is the wait for the bus; both vary. Is the total time $X+Y$ more or less variable than each part? What about the difference $X-Y$ ?

Section 2 of 14

Core idea

Start hereConverting units of a random quantity

Daily temperatures in °C have mean 20 and SD 5. In °F, $F=1.8C+32$ . What are the new mean and SD? Rules for $aX+b$ answer instantly: mean $68$ , SD $9$ .

The big idea

$E(aX+b)=aE(X)+b$ ; $\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$ . Adding shifts; multiplying scales the spread.

1. Proofs (discrete case)

X and Y = 2X + 3 for X with mean 5, variance 4

The

\times2

doubles the spread (s.d.

2\to4

) and the

+3

slides everything right. The variance goes from

4

16

E(aX+b)

and

\operatorname{Var}(aX+b)

$E(aX+b)=\sum(ax+b)p=a\sum xp+b\sum p=aE(X)+b$
Split; $\sum p=1$ .
$(aX+b)-E(aX+b)=a(X-\mu)$
Deviations scale by $a$ ; $b$ cancels.
$\operatorname{Var}(aX+b)=E[a^2(X-\mu)^2]=a^2\operatorname{Var}(X)$
Square.

Worked example $E(X)=5$ , $\operatorname{Var}(X)=4$

Find $E$ and Var of $2X+3$ and $3-X$ .

Goal: Values.
Strategy: Rules.

$E(2X+3)=13,\ \operatorname{Var}=16$
$2^2\times4$ .
$E(3-X)=-2,\ \operatorname{Var}=4$
$(-1)^2\times4$ .

Answer

13,16

;

-2,4

✓ CheckNegating never reduces spread ✓.

Common misconception $\operatorname{Var}(2X)\neq2\operatorname{Var}(X)$ It's

4\operatorname{Var}(X)

: variance is in squared units.

Pause and think

$\operatorname{Var}(X)=3$ . $\operatorname{Var}(4X-7)$ ?

Summary

Shift changes mean only.
Scale: mean × $a$ , variance × $a^2$ .

Linear transformation: $aX+b$ .

HistoryBienaymé's rule. The fact that variances of independent variables add was stated by the French statistician Irénée-Jules Bienaymé in 1853, and is sometimes called the Bienaymé formula. It is one reason variance, rather than standard deviation, became the standard measure of spread in mathematical statistics: standard deviations do not add, but variances do.

Fun factWhy spreads add in quadrature. Because variances add, standard deviations combine like the sides of a right-angled triangle:

\sigma_{X+Y}=\sqrt{\sigma_X^2+\sigma_Y^2}

. Engineers call this adding "in quadrature" when combining measurement errors. Two errors of

3

mm and

4

mm give a combined error of

5

mm, not

7

Section 3 of 14

Key formulae

One variable

E(aX+b)=aE(X)+b,\qquad\text{Var}(aX+b)=a^2\text{Var}(X)

Two variables (expectation, always)

E(aX+bY)=aE(X)+bE(Y)

Two independent variables (variance)

\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)

TrickSquare, then add. Always add. For a variance, square every coefficient (so every term becomes positive) and drop every constant.

\text{Var}(5-2X)=4\text{Var}(X)

;

\text{Var}(2X-3Y)=4\text{Var}(X)+9\text{Var}(Y)

. A minus sign in a variance answer is always a mistake.

Section 4 of 14

Worked examples

Example 1 · aX + b0 / 2 steps

E(X)=8

and

\text{Var}(X)=2

. Find

E(4X+1)

and

\text{Var}(4X+1)

Example 2 · A difference0 / 2 steps

X

and

Y

are independent:

E(X)=12

\text{Var}(X)=5

E(Y)=7

\text{Var}(Y)=3

. Find

E(2X-3Y)

and

\text{Var}(2X-3Y)

Example 3 · Changing units0 / 2 steps

Temperatures in °F have mean

68

and variance

81

. Find the mean and variance in °C, where

C=\dfrac{F-32}{1.8}

Section 5 of 14

Explore the graph

Standard deviations add like a right triangle

Loading the interactive graph…

Try this

The horizontal side is $\text{SD}(X)=a$ , the vertical side $\text{SD}(Y)=b$ . The hypotenuse is $\text{SD}(X\pm Y)=\sqrt{a^2+b^2}$ for independent $X$ , $Y$ .
With $a=3$ and $b=4$ , how long is the hypotenuse? Compare with $a+b$ .
Why is the combined SD always less than $a+b$ but more than either one?

Variances add; standard deviations combine by Pythagoras.

In the real worldInvestment portfolios. Splitting money equally between two independent investments, each with standard deviation

10\%

, gives a portfolio with

\text{Var}=\tfrac14(100)+\tfrac14(100)=50

, an SD of about

7.1\%

. The expected return is unchanged but the risk falls. This is the mathematics behind "don't put all your eggs in one basket".

Section 6 of 14

Common mistakes

Incorrect

\text{Var}(3X)=3\text{Var}(X)

Correct

9\text{Var}(X)

The coefficient is squared.

Incorrect

\text{Var}(X-Y)=\text{Var}(X)-\text{Var}(Y)

Correct

\text{Var}(X)+\text{Var}(Y)

(-1)^2=1

; uncertainties never cancel.

Incorrect

\text{Var}(X+5)=\text{Var}(X)+5

Correct

\text{Var}(X)

Shifting does not change spread.

IncorrectAdding standard deviations

CorrectAdd variances, then square-root

SDs combine in quadrature.

$\text{Var}(X-Y)=\text{Var}(X)-\text{Var}(Y)$

$\text{Var}(X-Y)=\text{Var}(X)+\text{Var}(Y)$

Section 7 of 14

Exam tip

Exam tipState "independent". When you add variances, write "since

X

and

Y

are independent". If a question asks for an assumption, independence is usually the one it wants, and the variance rule depends on it.

Section 8 of 14

Guided practice

SA-9709-P6-LC-G012 marksL1 FoundationCalculatoraX + b

E(X)=5

and

\text{Var}(X)=4

. Find

E(3X-2)

and

\text{Var}(3X-2)

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P6-LC1-0012 marksL1 FoundationCalculatoraX + b

E(X)=8

and

\text{Var}(X)=2

. Find

E(4X+1)

and

\text{Var}(4X+1)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-LC-0081 markL1 FoundationCalculatoraX + b

\text{Var}(X)=3

. Find

\text{Var}(5-2X)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-LC1-0021 markL1 FoundationCalculatoraX + b

The standard deviation of

X

3

. Find the standard deviation of

2X-5

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-LC-0012 marksL1 FoundationCalculatoraX + bY

X

and

Y

are independent with

E(X)=10

\text{Var}(X)=4

E(Y)=6

\text{Var}(Y)=9

. Find

E(X-Y)

and

\text{Var}(X-Y)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-LC1-0033 marksL2 StandardCalculatoraX + bY

X

and

Y

are independent with

E(X)=12

\text{Var}(X)=5

E(Y)=7

and

\text{Var}(Y)=3

. Find

E(2X-3Y)

and

\text{Var}(2X-3Y)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-LC-0112 marksL2 StandardCalculatoraX + bY

X

and

Y

are independent with

E(X)=4

\text{Var}(X)=1

E(Y)=3

\text{Var}(Y)=2

. Find

E(2X+3Y)

and

\text{Var}(2X+3Y)

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-LC1-0043 marksL3 ExamCalculatoraX + b

Temperatures

F

in °F have mean

68

and variance

81

. Find the mean and variance of the temperatures in °C, where

C=\dfrac{F-32}{1.8}

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$E(aX+b)=aE(X)+b$ ; $\text{Var}(aX+b)=a^2\text{Var}(X)$ .
$E(aX+bY)=aE(X)+bE(Y)$ .
Independent: $\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)$ .
Variances of sums and differences add.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P6-LC-Q011 markL1 FoundationCalculatoraX + b

\text{Var}(aX+b)

equals

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P6-LC-Q021 markL1 FoundationCalculatoraX + bY

For independent

X

and

Y

\text{Var}(X-Y)

equals

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-9709-P6-LC1-0051 markL1 FoundationCalculatoraX + bY

The result

\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)

requires that

Salam Academy Original Practice · Proposed mark scheme

Q4 · SA-9709-P6-LC-0071 markL1 FoundationCalculatoraX + bY

X\sim\text{Po}(2)

. Which of these has a Poisson distribution?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics
Where: WST03 (Statistics 3) · Combinations of random variables
Expected: E(aX ± bY) and Var(aX ± bY) for independent random variables.
Calculator: Permitted. Statistical tables are provided in the formula booklet.
Typical style: Often a packaging context: the total mass of several items plus a box, compared with a limit.
Open in: YMA01 · WST03 9FM0 · Option

Section 14 of 14

Past paper questions

1 question (6 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 26 marks
Standard deviations of combinations of normal and Poisson variables.
Question paper, page 3 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).