Linear combinations of random variables

CIE A-Level Mathematics· 9709 Unit 4: Probability & Statistics 1 & 2, Section 8· 25 min read

1. Linearity of Expectation for Linear Combinations★★☆☆☆⏱ 6 min

📘 Definition

Linearity of Expectation

$E [a X + bY + c] = a E [X] + b E [Y] + c$

This identity holds for any pair of random variables, no matter if they are dependent or independent, for all real constants a, b, c.

This rule extends naturally to any number of variables. For a sum of $n$ random variables $X_{1}, X_{2}, ... X_{n}$ with weights $a_{1}, a_{2} ... a_{n}$ , the expected value is $E [\sum a_{i} X_{i}] = \sum a_{i} E [X_{i}]$ . No additional conditions apply, making this one of the most robust identities in probability theory.

📐 Worked Example

Given $E [X] = 5$ , $E [Y] = 3$ , calculate the value of $E [4 X - 2 Y + 7]$

1
Apply linearity of expectation directly, no independence checks required
2
$E[4X - 2Y +7] = 4E[X] - 2E[Y] +7$
3
$= 4(5) - 2(3) +7 = 20 -6 +7 = 21$

✓ Quick check

Test your understanding of linearity of expectation:

Which of these values is equal to $E [2 A + 3 B - 4]$ if $E [A] = 2$ , $E [B] = 1$ ?
- 3
- 5
- 7
- 9
Reveal answer
3 —
Calculation: 22 + 31 -4 = 4 +3 -4 = 3, no dependence between A and B affects the result.

Exam tip:

CIE exam mark schemes award full 2/3 marks for correct substitution of values into the linearity of expectation formula, no working beyond the final calculation is required unless stated.

2. Variance of Linear Combinations★★★☆☆⏱ 7 min

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

If X and Y are independent, their covariance $C o v (X, Y) = 0$ , so the formula simplifies to $V a r (a X + bY) = a^{2} V a r (X) + b^{2} V a r (Y)$ . Note that variance never scales linearly with constants: all coefficients are squared, and the constant offset $c$ added to the combination has no impact on variance, as it does not change the spread of the distribution.

📐 Worked Example

Given $V a r (X) = 9$ , $V a r (Y) = 16$ , X and Y are independent. Calculate $V a r (3 X - 2 Y + 10)$

1
Drop the constant offset 10 first, as it does not affect variance
2
Square all coefficients, sum the individual variances multiplied by their squared coefficients
3
$Var(3X -2Y) = 3^2 Var(X) + (-2)^2 Var(Y)$
4
$= 9*9 + 4*16 = 81 + 64 = 145$

Exam tip:

You must explicitly state 'X and Y are independent' in your working to justify dropping the covariance term, to avoid losing 1 mark in the exam.

3. Linear Combinations of Independent Normal Variables★★★☆☆⏱ 6 min

✓ Calculator OK

📘 Definition

Reproductive Property of Normal Distribution

Any linear combination of independent normally distributed random variables is also normally distributed.

If $X \sim N (μ_{X}, σ_{X}^{2})$ and $Y \sim N (μ_{Y}, σ_{Y}^{2})$ and X and Y are independent, then $a X + bY + c \sim N (a μ_{X} + b μ_{Y} + c, a^{2} σ_{X}^{2} + b^{2} σ_{Y}^{2})$ . This property is unique to a small set of common distributions, and is tested frequently in CIE Paper 6 and Paper 7.

📐 Worked Example

The mass of a small apple $A \sim N (120, 1 5^{2})$ g, the mass of a banana $B \sim N (150, 2 0^{2})$ g. Find the probability that the total mass of 1 apple and 1 banana is less than 300g.

1
Define the total mass $T = A + B$
2
$E[T] = 120 + 150 = 270$
3
$Var(T) = 15^2 + 20^2 = 225 + 400 = 625, \sigma_T = 25$
4
So $T \sim N (270, 2 5^{2})$
5
$P(T < 300) = \Phi\left(\frac{300 - 270}{25}\right) = \Phi(1.2) = 0.8849$

4. Distinguishing $aX$ vs $X_1 + X_2 + ... + X_a$★★★★☆⏱ 6 min

This is the single most commonly tested trick in this topic. If you take a single random variable X and multiply it by 2, you get $2 X$ , with variance $4 V a r (X)$ . If you take two independent copies of X, $X_{1}$ and $X_{2}$ , each with the same distribution as X, their sum $X_{1} + X_{2}$ has variance $V a r (X_{1}) + V a r (X_{2}) = 2 V a r (X)$ . These are completely different distributions, and exam questions explicitly test this distinction.

Methods compared

Side by side comparison of the two cases:

Single variable multiplied by 2: $2X$

Every value of X is doubled, so the spread of the distribution is doubled

+ Pros: Simple scaling of the original distribution

− Cons: Variables are perfectly correlated, not independent

Sum of 2 independent copies: $X_1 + X_2$

Two separate independent observations from the same distribution, summed

+ Pros: No covariance term, variance adds linearly

− Cons: Distribution shape is different from scaled X

📐 Worked Example

Given $X \sim N (10, 4)$ , find the difference between $V a r (3 X)$ and $V a r (X_{1} + X_{2} + X_{3})$ where $X_{1}, X_{2}, X_{3}$ are independent copies of X.

1
$Var(3X) = 3^2 Var(X) = 9 * 4 = 36$
2
$Var(X_1 + X_2 + X_3) = Var(X) + Var(X) + Var(X) = 4 + 4 + 4 = 12$
3
Difference = 36 - 12 = 24

Exam tip:

Read the question wording carefully: 'the total of 3 independent observations' means sum of 3 variables, 'three times the value of a single observation' means 3X.

5. Common Pitfalls

Wrong move:

Calculating $V a r (a X)$ as $aV a r (X)$ instead of $a^{2} V a r (X)$

Why:

Forgetting variance measures squared spread, so scaling a variable by a scales its variance by $a^{2}$

Correct move:

Always square the constant coefficient when calculating variance of a scaled variable

Wrong move:

Calculating $V a r (X - Y)$ as $V a r (X) - V a r (Y)$

Why:

Treating variance like expectation, where the minus sign carries through

Correct move:

Square the negative coefficient, so $V a r (X - Y) = V a r (X) + V a r (Y)$ for independent variables

Wrong move:

Treating $2 X$ as equivalent to $X_{1} + X_{2}$

Why:

Confusing scaling a single variable with summing two independent copies

Correct move:

Check question wording: if it refers to multiple separate observations, sum their variances directly

Wrong move:

Including constant offsets in variance calculations

Why:

Assuming adding a fixed number to a distribution changes its spread

Correct move:

Ignore all constant terms when calculating variance, they only affect the expected value

Wrong move:

Applying the normal reproductive property to dependent variables

Why:

Forgetting the rule only applies to independent normal variables

Correct move:

Explicitly confirm independence is stated in the question before using the simplified variance formula

6. Quick Reference Cheatsheet

Operation	Expectation	Variance (independent variables)	Variance (dependent variables)
$a X + b$	$a E [X] + b$	$a^{2} V a r (X)$	$a^{2} V a r (X)$
$a X + bY$	$a E [X] + b E [Y]$	$a^{2} V a r (X) + b^{2} V a r (Y)$	$a^{2} V a r (X) + b^{2} V a r (Y) + 2 ab C o v (X, Y)$
$X_{1} + X_{2} + ... + X_{n}$	$\sum E [X_{i}]$	$\sum V a r (X_{i})$	$\sum V a r (X_{i}) + 2 \sum_{i < j} C o v (X_{i}, X_{j})$

7. Frequently Asked

Do I need variables to be independent to calculate the expected value of a linear combination?

No. The linearity of expectation holds for all random variables, regardless of dependence. Independence is only required to simplify the variance calculation to remove covariance terms.

Why is the sum of two normal variables also normal?

The normal distribution is closed under linear transformations, meaning any linear combination of independent normally distributed variables will itself follow a normal distribution, with parameters derived from the sum of individual expectations and variances.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 7
Sum of 3 independent test scores
2022 · 6
Difference of two normal variables
2021 · 7
Var(2X) vs Var(X1+X2) comparison

Going deeper

official_syllabusCIE 9709 Formula SheetReference for standard expectation and variance identities

What's Next

Mastering linear combinations of random variables is the critical foundation for sampling distributions, where you will analyze the behaviour of sums and means of large sets of independent observations. This topic also directly supports hypothesis testing for population means, and linear regression analysis, where you will model observed values as linear combinations of parameters and error terms. You will see this concept reused in almost all subsequent Probability & Statistics 2 topics, so ensure you can reliably distinguish between scaled single variables and sums of independent copies to avoid losing easy marks in your exam.