What is a linear combination of random variables in the context of Cambridge International A Levels Mathematics?

In Cambridge International A Levels Mathematics, a linear combination of random variables refers to an expression formed by multiplying each random variable by a constant and then summing the results. For example, if X and Y are random variables, a linear combination could be expressed as aX + bY, where a and b are constants. This concept is crucial in understanding how different random variables can be combined and analyzed together.

How do you calculate the expected value of a linear combination of random variables?

To calculate the expected value of a linear combination of random variables, you apply the linearity of expectation. If you have a linear combination aX + bY, where X and Y are random variables and a and b are constants, the expected value is E(aX + bY) = aE(X) + bE(Y). This property simplifies the process of finding expected values in Probability and Statistics 2.

What is the variance of a linear combination of independent random variables?

For independent random variables X and Y, the variance of a linear combination aX + bY is calculated using the formula Var(aX + bY) = a²Var(X) + b²Var(Y). This formula assumes that X and Y are independent, which is a common scenario in Cambridge International A Levels Probability and Statistics 2. It highlights how the variance of a linear combination depends on the variances of the individual random variables and the constants involved.

Why is understanding linear combinations of random variables important in Probability and Statistics 2?

Understanding linear combinations of random variables is crucial in Probability and Statistics 2 because it allows students to analyze and predict outcomes when multiple random variables are involved. This concept is fundamental in fields such as finance, engineering, and data science, where decisions are often based on the combined effect of several random factors. Mastery of this topic helps students develop a deeper understanding of statistical modeling and inference.

Can you provide an example of a real-world application of linear combinations of random variables?

A real-world application of linear combinations of random variables is in portfolio management in finance. Here, the return on a portfolio is a linear combination of the returns on individual assets. By analyzing these combinations, investors can optimize their portfolios to achieve desired risk-return profiles. This application demonstrates the practical importance of understanding linear combinations in making informed financial decisions, a topic covered in Cambridge International A Levels Mathematics.

Why is "Subtracting variance for $X - Y$" a common mistake in Linear combinations of random variables?

Why it happens: Mimicking expectation. How to avoid it: Var(X - Y) = Var(X) + Var(Y) (for independent). Sign DOESN'T affect variance. Source: 9709 Examiner Reports 2022-2024

Why is "Assuming independence when not stated" a common mistake in Linear combinations of random variables?

Why it happens: Skipping conditions. How to avoid it: Verify question states independence (or context implies it). Otherwise variance formula fails. Source: 9709 Examiner Reports 2022-2024

Cambridge International A Levels Mathematics (9709)

Linear combinations of random variables

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Short Notes - Linear combinations of random variables

Detailed Study Notes

Detailed notes on Probability and Statistics 2 - Paper 6 for Cambridge International A Levels Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Linear Combinations of Random Variables Study Notes — Cambridge International A Level Mathematics 9709 S2 (2024-2026 syllabus)

$E$ and $\text{Var}$ of $aX + b$ and $aX + bY$ . Independent normals stay normal.

At a glance

$E(aX + b) = aE(X) + b$ .
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ .
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (independent).
Sign doesn't affect variance.
Normals stay normal under linear combinations.

What you’ll learn

Mapped to the Cambridge International A Level 9709 syllabus (2024-2026).

S2.2.1 — Compute $E$ and $\text{Var}$ of linear transformations.
S2.2.2 — Compute for sums and differences (independent).
S2.2.3 — Distribution of linear combinations of normals.

Linear transformations of one RV

$aX + b$ .

Expectation. $E(aX + b) = aE(X) + b$ .

Variance. $\text{Var}(aX + b) = a^2 \text{Var}(X)$ .

Note. Adding constant $b$ DOESN'T affect variance (just shifts mean).

Scaling. Multiplying by $a$ scales variance by $a^2$ . So $\sigma$ scales by $|a|$ .

Example. $E(X) = 5$ , $\text{Var}(X) = 4$ . Find $E$ and $\text{Var}$ of $2X + 3$ .

$E(2X + 3) = 2(5) + 3 = 13$ .
$\text{Var}(2X + 3) = 4 \cdot 4 = 16$ .

Cambridge tip. Always square the coefficient for variance.

Adding a constant shifts every value equally, so the spread cannot change. Multiplying stretches the gaps, and variance being a squared measure picks up a², not a.

$E(aX + b) = aE(X) + b$ .
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ .
Constant doesn't affect variance.

See the full worked example for linear combinations of random variables →

Sums and differences

Independent RVs.

Linearity of expectation. Always holds: $E(aX + bY) = aE(X) + bE(Y)$ .

Variance (independence required). $\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$ .

Key insight. $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (signs squared, so both positive).

More than two. $\text{Var}\left(\sum a_i X_i\right) = \sum a_i^2 \text{Var}(X_i)$ for independent $X_i$ .

Example. $E(X) = 3$ , $\text{Var}(X) = 1$ , $E(Y) = 4$ , $\text{Var}(Y) = 2$ .

$E(2X - Y) = 6 - 4 = 2$ .
$\text{Var}(2X - Y) = 4(1) + 1(2) = 6$ .

Cambridge tip. State independence explicitly; required for variance formula.

Means combine with the signs you wrote down. Variances do not: every coefficient is squared, so independent variances always ADD however the combination is written.

Var(X − Y) = Var(X) + Var(Y) for independent X and Y. Subtracting removes the means, not the uncertainty, so the difference is more variable than either variable alone.

Linearity for $E$ .
Variance: squared coefficients.
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ .

See the full worked example for linear combinations of random variables →

Linear combinations of normals

Stay normal.

Key result. If $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$ are INDEPENDENT, then $aX + bY \sim N(a\mu_1 + b\mu_2, a^2\sigma_1^2 + b^2\sigma_2^2)$ .

Special cases.

$X + Y \sim N(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)$ .
$X - Y \sim N(\mu_1 - \mu_2, \sigma_1^2 + \sigma_2^2)$ .

Example. $X \sim N(50, 16)$ , $Y \sim N(40, 9)$ .

$X - Y \sim N(10, 25)$ .

Sample mean. If $X_1, \ldots, X_n$ are iid $N(\mu, \sigma^2)$ : $\bar{X} \sim N(\mu, \sigma^2/n)$ .

Cambridge tip. Always state independence and that normals are involved.

Any linear combination of independent normals is itself normal, so you only need its mean and variance. Note the sum and the difference have the SAME variance, hence the same width.

3X multiplies a single observation, so its variance carries a factor 3². X₁ + X₂ + X₃ adds three independent observations, so the variances simply add. Read the wording carefully.

Linear combo of independent normals is normal.
Mean and variance via standard formulas.
$\bar{X}$ for sample mean.

See the full worked example for linear combinations of random variables →

Quick recap

$E(aX + b) = aE(X) + b$ .
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ .
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (independent).
Linear combos of independent normals are normal.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

$E$ formulas.
$\text{Var}$ formulas (squared coefficients).
Independence required for sum variance.
Normal stays normal.

How it’s examined

Linear combinations appear every S2 — typically 8-12 marks. Most-tested: sum/difference of normals (7 marks), linear transformation (5 marks), sample mean (5-7 marks).

Sources: Cambridge International A Level Mathematics 9709 syllabus (2024-2026); 9709 Examiner Reports 2022-2024; 9709/62 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

Take this whole topic with you

Step-by-step worked examples — Linear combinations of random variables

Step-by-step solutions to past-paper-style questions on linear combinations of random variables, written exactly the way a tutor would explain them at the board.

1 $E(aX + b)$ , $\text{Var}(aX + b)$ (5 marks)
Extended• Adapted from 9709/62 May/Jun 2024• linear
Question
$X$ has $E(X) = 5$ , $\text{Var}(X) = 4$ . Find $E(2X + 3)$ and $\text{Var}(2X + 3)$ . (5 marks)
Step-by-step solution
1. Step 1
  $E(aX + b) = aE(X) + b$ .
  $E(2X + 3) = 2(5) + 3 = 13$
2. Step 2
  $\text{Var}(aX + b) = a^2 \text{Var}(X)$ (constant shift doesn't affect variance).
  $\text{Var}(2X + 3) = 4 \times 4 = 16$
Answer
$E = 13$ ; $\text{Var} = 16$ .
2Sum of independent variables (7 marks)
Extended• sum
Question
$X$ has $E(X) = 3$ , $\text{Var}(X) = 1$ . $Y$ has $E(Y) = 4$ , $\text{Var}(Y) = 2$ . $X$ and $Y$ independent. Find $E(2X - Y)$ and $\text{Var}(2X - Y)$ . (7 marks)
Step-by-step solution
1. Step 1
  Linearity of expectation.
  $E(2X - Y) = 2E(X) - E(Y) = 6 - 4 = 2$
2. Step 2
  Variance of independent sum. $\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$ .
3. Step 3
  Note: signs squared. $\text{Var}(2X - Y) = 4 \cdot 1 + 1 \cdot 2 = 6$ .
Answer
$E = 2$ ; $\text{Var} = 6$ .
3Linear combination of normals (7 marks)
Extended• normal combination
Question
$X \sim N(50, 16)$ , $Y \sim N(40, 9)$ independent. Find distribution of $X - Y$ . (7 marks)
Step-by-step solution
1. Step 1
  Linear combination of independent normals is normal.
2. Step 2
  Mean. $E(X - Y) = 50 - 40 = 10$ .
3. Step 3
  Variance. $\text{Var}(X - Y) = 16 + 9 = 25$ (signs squared, so $+$ either way).
4. Step 4
  Distribution.
  $X - Y \sim N(10, 25)$
Answer
$X - Y \sim N(10, 25)$ .

Key Formulae — Linear combinations of random variables

The formulae you need to memorise for linear combinations of random variables on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.

Linear expectation
$E(aX + b) = aE(X) + b; \quad E(aX + bY) = aE(X) + bE(Y)$
When to use
Linearity always holds.
Variance of linear combination
$\text{Var}(aX + b) = a^2 \text{Var}(X); \quad \text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$
When to use
Variance scales by squared coefficient. Independence required for sum (second formula).
Normal combination
$aX + bY \sim N(aE(X) + bE(Y),\; a^2\text{Var}(X) + b^2\text{Var}(Y))$
When to use
For independent normals $X, Y$ , any linear combination is also normal. Variance uses squared coefficients.

Key Definitions and Keywords — Linear combinations of random variables

Definitions to memorise and the exact keywords mark schemes credit for linear combinations of random variables answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.

Linear combination
Examiner keyword
Expression of form $aX + bY + \ldots$ with constants $a, b, \ldots$
Independent random variables
Examiner keyword
Outcomes don't affect each other. Required for variance of sum formula.

Common Mistakes and Misconceptions — Linear combinations of random variables

The traps other students keep falling into on linear combinations of random variables questions — taken from recent Cambridge International A Level 9709 examiner reports and mark schemes — and how to avoid them.

✕Subtracting variance for $X - Y$
9709 Examiner Reports 2022-2024
Why it happens
Mimicking expectation.
How to avoid it
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (for independent). Sign DOESN'T affect variance.
✕Assuming independence when not stated
9709 Examiner Reports 2022-2024
Why it happens
Skipping conditions.
How to avoid it
Verify question states independence (or context implies it). Otherwise variance formula fails.

Linear combinations of random variables — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Cambridge International A Levels Mathematics (9709)

Linear combinations of random variables

0% Complete

3 tasks remaining

Short Notes - Linear combinations of random variables

Detailed Study Notes

Detailed notes on Probability and Statistics 2 - Paper 6 for Cambridge International A Levels Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Linear Combinations of Random Variables Study Notes — Cambridge International A Level Mathematics 9709 S2 (2024-2026 syllabus)

$E$ and $\text{Var}$ of $aX + b$ and $aX + bY$ . Independent normals stay normal.

At a glance

$E(aX + b) = aE(X) + b$ .
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ .
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (independent).
Sign doesn't affect variance.
Normals stay normal under linear combinations.

What you’ll learn

Mapped to the Cambridge International A Level 9709 syllabus (2024-2026).

S2.2.1 — Compute $E$ and $\text{Var}$ of linear transformations.
S2.2.2 — Compute for sums and differences (independent).
S2.2.3 — Distribution of linear combinations of normals.

Linear transformations of one RV

$aX + b$ .

Expectation. $E(aX + b) = aE(X) + b$ .

Variance. $\text{Var}(aX + b) = a^2 \text{Var}(X)$ .

Note. Adding constant $b$ DOESN'T affect variance (just shifts mean).

Scaling. Multiplying by $a$ scales variance by $a^2$ . So $\sigma$ scales by $|a|$ .

Example. $E(X) = 5$ , $\text{Var}(X) = 4$ . Find $E$ and $\text{Var}$ of $2X + 3$ .

$E(2X + 3) = 2(5) + 3 = 13$ .
$\text{Var}(2X + 3) = 4 \cdot 4 = 16$ .

Cambridge tip. Always square the coefficient for variance.

Adding a constant shifts every value equally, so the spread cannot change. Multiplying stretches the gaps, and variance being a squared measure picks up a², not a.

$E(aX + b) = aE(X) + b$ .
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ .
Constant doesn't affect variance.

See the full worked example for linear combinations of random variables →

Sums and differences

Independent RVs.

Linearity of expectation. Always holds: $E(aX + bY) = aE(X) + bE(Y)$ .

Variance (independence required). $\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$ .

Key insight. $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (signs squared, so both positive).

More than two. $\text{Var}\left(\sum a_i X_i\right) = \sum a_i^2 \text{Var}(X_i)$ for independent $X_i$ .

Example. $E(X) = 3$ , $\text{Var}(X) = 1$ , $E(Y) = 4$ , $\text{Var}(Y) = 2$ .

$E(2X - Y) = 6 - 4 = 2$ .
$\text{Var}(2X - Y) = 4(1) + 1(2) = 6$ .

Cambridge tip. State independence explicitly; required for variance formula.

Means combine with the signs you wrote down. Variances do not: every coefficient is squared, so independent variances always ADD however the combination is written.

Var(X − Y) = Var(X) + Var(Y) for independent X and Y. Subtracting removes the means, not the uncertainty, so the difference is more variable than either variable alone.

Linearity for $E$ .
Variance: squared coefficients.
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ .

See the full worked example for linear combinations of random variables →

Linear combinations of normals

Stay normal.

Key result. If $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$ are INDEPENDENT, then $aX + bY \sim N(a\mu_1 + b\mu_2, a^2\sigma_1^2 + b^2\sigma_2^2)$ .

Special cases.

$X + Y \sim N(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2)$ .
$X - Y \sim N(\mu_1 - \mu_2, \sigma_1^2 + \sigma_2^2)$ .

Example. $X \sim N(50, 16)$ , $Y \sim N(40, 9)$ .

$X - Y \sim N(10, 25)$ .

Sample mean. If $X_1, \ldots, X_n$ are iid $N(\mu, \sigma^2)$ : $\bar{X} \sim N(\mu, \sigma^2/n)$ .

Cambridge tip. Always state independence and that normals are involved.

Any linear combination of independent normals is itself normal, so you only need its mean and variance. Note the sum and the difference have the SAME variance, hence the same width.

3X multiplies a single observation, so its variance carries a factor 3². X₁ + X₂ + X₃ adds three independent observations, so the variances simply add. Read the wording carefully.

Linear combo of independent normals is normal.
Mean and variance via standard formulas.
$\bar{X}$ for sample mean.

See the full worked example for linear combinations of random variables →

Quick recap

$E(aX + b) = aE(X) + b$ .
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ .
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (independent).
Linear combos of independent normals are normal.

Memorise this

Verbatim phrases and definitions Cambridge mark schemes credit.

$E$ formulas.
$\text{Var}$ formulas (squared coefficients).
Independence required for sum variance.
Normal stays normal.

How it’s examined

Linear combinations appear every S2 — typically 8-12 marks. Most-tested: sum/difference of normals (7 marks), linear transformation (5 marks), sample mean (5-7 marks).

Sources: Cambridge International A Level Mathematics 9709 syllabus (2024-2026); 9709 Examiner Reports 2022-2024; 9709/62 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

Take this whole topic with you

Step-by-step worked examples — Linear combinations of random variables

Step-by-step solutions to past-paper-style questions on linear combinations of random variables, written exactly the way a tutor would explain them at the board.

1 $E(aX + b)$ , $\text{Var}(aX + b)$ (5 marks)
Extended• Adapted from 9709/62 May/Jun 2024• linear
Question
$X$ has $E(X) = 5$ , $\text{Var}(X) = 4$ . Find $E(2X + 3)$ and $\text{Var}(2X + 3)$ . (5 marks)
Step-by-step solution
1. Step 1
  $E(aX + b) = aE(X) + b$ .
  $E(2X + 3) = 2(5) + 3 = 13$
2. Step 2
  $\text{Var}(aX + b) = a^2 \text{Var}(X)$ (constant shift doesn't affect variance).
  $\text{Var}(2X + 3) = 4 \times 4 = 16$
Answer
$E = 13$ ; $\text{Var} = 16$ .
2Sum of independent variables (7 marks)
Extended• sum
Question
$X$ has $E(X) = 3$ , $\text{Var}(X) = 1$ . $Y$ has $E(Y) = 4$ , $\text{Var}(Y) = 2$ . $X$ and $Y$ independent. Find $E(2X - Y)$ and $\text{Var}(2X - Y)$ . (7 marks)
Step-by-step solution
1. Step 1
  Linearity of expectation.
  $E(2X - Y) = 2E(X) - E(Y) = 6 - 4 = 2$
2. Step 2
  Variance of independent sum. $\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$ .
3. Step 3
  Note: signs squared. $\text{Var}(2X - Y) = 4 \cdot 1 + 1 \cdot 2 = 6$ .
Answer
$E = 2$ ; $\text{Var} = 6$ .
3Linear combination of normals (7 marks)
Extended• normal combination
Question
$X \sim N(50, 16)$ , $Y \sim N(40, 9)$ independent. Find distribution of $X - Y$ . (7 marks)
Step-by-step solution
1. Step 1
  Linear combination of independent normals is normal.
2. Step 2
  Mean. $E(X - Y) = 50 - 40 = 10$ .
3. Step 3
  Variance. $\text{Var}(X - Y) = 16 + 9 = 25$ (signs squared, so $+$ either way).
4. Step 4
  Distribution.
  $X - Y \sim N(10, 25)$
Answer
$X - Y \sim N(10, 25)$ .

Key Formulae — Linear combinations of random variables

Linear expectation
$E(aX + b) = aE(X) + b; \quad E(aX + bY) = aE(X) + bE(Y)$
When to use
Linearity always holds.
Variance of linear combination
$\text{Var}(aX + b) = a^2 \text{Var}(X); \quad \text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$
When to use
Variance scales by squared coefficient. Independence required for sum (second formula).
Normal combination
$aX + bY \sim N(aE(X) + bE(Y),\; a^2\text{Var}(X) + b^2\text{Var}(Y))$
When to use
For independent normals $X, Y$ , any linear combination is also normal. Variance uses squared coefficients.

Key Definitions and Keywords — Linear combinations of random variables

Linear combination
Examiner keyword
Expression of form $aX + bY + \ldots$ with constants $a, b, \ldots$
Independent random variables
Examiner keyword
Outcomes don't affect each other. Required for variance of sum formula.

Common Mistakes and Misconceptions — Linear combinations of random variables

✕Subtracting variance for $X - Y$
9709 Examiner Reports 2022-2024
Why it happens
Mimicking expectation.
How to avoid it
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (for independent). Sign DOESN'T affect variance.
✕Assuming independence when not stated
9709 Examiner Reports 2022-2024
Why it happens
Skipping conditions.
How to avoid it
Verify question states independence (or context implies it). Otherwise variance formula fails.

Linear combinations of random variables — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Linear combinations of random variables

Short Notes - Linear combinations of random variables

Detailed Study Notes

At a glance

What you’ll learn

Linear transformations of one RV

Sums and differences

Linear combinations of normals

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1E(aX+b)E(aX + b)E(aX+b), Var(aX+b)\text{Var}(aX + b)Var(aX+b) (5 marks)

2Sum of independent variables (7 marks)

3Linear combination of normals (7 marks)

Linear expectation

Variance of linear combination

Normal combination

Linear combination

Independent random variables

✕Subtracting variance for X−YX - YX−Y

✕Assuming independence when not stated

Linear combinations of random variables — frequently asked questions

Linear combinations of random variables

Short Notes - Linear combinations of random variables

Detailed Study Notes

At a glance

What you’ll learn

Linear transformations of one RV

Sums and differences

Linear combinations of normals

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1E(aX+b)E(aX + b)E(aX+b), Var(aX+b)\text{Var}(aX + b)Var(aX+b) (5 marks)

2Sum of independent variables (7 marks)

3Linear combination of normals (7 marks)

Linear expectation

Variance of linear combination

Normal combination

Linear combination

Independent random variables

✕Subtracting variance for X−YX - YX−Y

✕Assuming independence when not stated

Linear combinations of random variables — frequently asked questions

1 $E(aX + b)$ , $\text{Var}(aX + b)$ (5 marks)

✕Subtracting variance for $X - Y$

1 $E(aX + b)$ , $\text{Var}(aX + b)$ (5 marks)

✕Subtracting variance for $X - Y$