Learn9709Paper 6 · Probability & Statistics 2Linear combinations of random variablesSums of observations and multiples

Linear combinations of random variables · Subtopic 2 of 3

Sums of Observations versus Multiples

Weighing one bag and multiplying by $4$ is not the same as weighing four different bags. Both totals have the same mean, but $4X$ has variance $16\sigma^2$ while $X_1+X_2+X_3+X_4$ has only $4\sigma^2$ .

Cambridge 9709 Paper 6Edexcel YMA01 WST03Edexcel 9FM0 Option≈ 35 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Distinguish $nX$ (one value scaled) from $X_1+\cdots+X_n$ (independent values added).
Use $\text{Var}(nX)=n^2\sigma^2$ and $\text{Var}(X_1+\cdots+X_n)=n\sigma^2$ .
Find probabilities for totals of normal observations.
Find the distribution of the mean of $n$ observations.

Pause and think

Bags of rice vary in mass. Which total is more likely to be far from its mean: four times the mass of one bag, or the combined mass of four different bags?

Section 2 of 14

Core idea

Start hereTwo apples or one big apple?

Apples weigh 100 g on average with variance 25. Two random apples together: mean 200 g. One apple's weight doubled: also mean 200 g. But the two apples' total is less variable: one light apple tends to be balanced by the other.

The big idea

For independent $X$ , $Y$ : $E(X\pm Y)=E(X)\pm E(Y)$ and $\operatorname{Var}(X\pm Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$ . Note $\operatorname{Var}(X_1+X_2)=2\operatorname{Var}(X)$ but $\operatorname{Var}(2X)=4\operatorname{Var}(X)$ .

1. Sum vs multiple

X₁ + X₂ versus 2X (mean 100, variance 25 per apple)

Same centre, different spread. Adding independent values partly cancels out their randomness.

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

$\pm$: variances add even for a difference, since $(-b)^2=b^2$ .

Worked exampleBag of 6 apples

Apples $\sim N(100,25)$ . P(total of 6 > 620 g)?

Goal: Probability.
Strategy: Sum of 6 independent normals.

$T\sim N(600,150)$
$6\times100$ ; $6\times25$ .
$z=\tfrac{20}{\sqrt{150}}=1.633$
Standardise.
$1-0.9488=0.0512$
Upper tail.

Answer

0.0512

✓ CheckUsing

6X

wrongly: variance

900

z=0.667

0.25

: far too large ✓.

Worked example $2X-3Y$

$X$ : mean 10, var 4. $Y$ : mean 5, var 1. Find mean and variance of $2X-3Y$ .

Goal: Values.
Strategy: Rules.

$E=20-15=5$
Means subtract.
$\operatorname{Var}=4(4)+9(1)=25$
Squares of coefficients, added.

AnswerMean

5

, variance

25

✓ CheckSD

5

✓.

Pause and think

Var(X − Y) if both have variance 9?

Summary

Variances always add.
$\sum X_i\neq nX$ in variance.

Independent: Values do not affect each other.

HistoryThe square-root law. Because variances of independent sums grow like

n

, the spread of a total grows only like

\sqrt n

. Abraham de Moivre noticed this in 1730 and called it the most important rule in the theory of chance. It explains why averaging many measurements makes them more precise.

Fun factThe lift sign. A lift sign says "

8

persons,

640

kg". That assumes an average of

80

kg, but with real, independent people the total varies. Engineers build in a big safety margin, because the chance that

8

random adults exceed

700

kg is small but not zero (see the practice questions).

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