AHL 4.14 — Linear combinations of random variables (expectation & variance)

This section explains how expectations and variances behave under linear transformations and linear combinations.
You will learn formulas for E(aX + b), Var(aX + b), and for sums of independent variables, plus why
the sample mean and the sample variance (with n−1) are unbiased estimators. Emphasis is on interpretation and use.

Term / concept Definition / short explanation
Expectation / Mean (E) The long-run average value of a random variable. Notation: E(X) = μ.
Variance (Var) Expected squared deviation from the mean: Var(X) = E[(X − μ)2] = σ2.
Linear transform Y = aX + b. Scale by a, shift by b. Affects mean and variance in specific ways (below).
Sample mean & variance \u0305X = (1/n) Σ xi; s2n−1 = (1/(n−1)) Σ (xi − \u0305X)2 — unbiased estimates of μ and σ2.

📌 1. Expectation under linear transformation

If Y = aX + b (a and b constants) then:

E(aX + b) = a E(X) + b.

  • Why: expectation is linear — sum and constants can be pulled out of E(·).
  • Interpretation: scaling X by a multiplies the mean by a; shifting by b adds b to the mean.
  • Example (font Times Newer Roman):
    If E(X) = 10 and Y = 3X − 4 then E(Y) = 3×10 − 4 = 26.

📐 IA spotlight

When creating an IA where you transform data, explicitly compute E before and after transformation to show understanding.
If you scale raw measurements (e.g., convert metres to centimetres), show how the mean scales accordingly.

📌 2. Variance under linear transformation

For Y = aX + b:

Var(aX + b) = a2 Var(X).

  • Why b disappears: adding a constant shifts every observation by the same amount — it does not change spread.
  • Effect of a: multiplying by a scales spread by |a|, and variance (which squares deviations) scales by a2.
  • Example (Times Newer Roman):
    If Var(X) = 4 and Y = −2X + 5 then Var(Y) = (−2)2 × 4 = 4 × 4 = 16.

🧠 Examiner tip

  • When asked for Var(aX + b) show the substitution and explicitly state Var(aX + b) = a2Var(X). If given numbers, compute both steps (square a, multiply by given variance).
  • Always comment in words: “variance increases (or decreases) by factor a2.”

📌 3. Expectation and variance of linear combinations (several variables)

Let X1, X2, …, Xn be random variables and ai constants. For the linear combination
S = Σ ai Xi:

  • Expectation (always): E(S) = Σ ai E(Xi).
  • Variance (if Xi are independent): Var(S) = Σ ai2 Var(Xi).
  • If not independent: covariances appear: Var(S) = Σ ai2Var(Xi) + 2 Σi<j aiaj Cov(Xi,Xj).
  • Practical point: Many exam problems assume independence so the simpler sum-of-variances formula applies.

Example (font Times Newer Roman):

Let X and Y be independent with Var(X)=9, Var(Y)=4. For Z = 2X − 3Y:
Var(Z) = 22×9 + (−3)2×4 = 4×9 + 9×4 = 36 + 36 = 72.

🌍 Real-world connection

Portfolio variance in finance is computed by combining variances of asset returns and their covariances.
Independence is rare — covariances matter. This shows why the “sum of variances” formula must be used carefully in applications.

📌 4. Sample mean and unbiasedness

Given a random sample X1, …, Xn from a population with mean μ and variance σ2:

  • Sample mean: SX = (1/n) Σ si. It is an unbiased estimator of μ:
    E(x̄) = μ.
  • Variance of sample mean (if Xi independent):
    Var(x̄) = σ2 / n. So averaging reduces variance by factor n.

Example:

Population σ2 = 16, n = 25 → Var(x̄) = 16 / 25 = 0.64. The standard error = √0.64 = 0.8.

📌 5. Unbiased sample variance (s2n−1)

The sample variance with denominator (n−1) is:

s2n−1 = (1/(n−1)) Σ (xi − x̄)2

  • Unbiasedness: E[s2n−1] = σ2. Using n in denominator would systematically underestimate σ2.
  • Why n−1? Because we used the sample mean x̄ (an estimate) when computing deviations — one degree of freedom is lost.
  • Classroom check: for grouped data replace sums by Σ fi(xi − x̄)2 and use n = Σ fi.

🔍 TOK perspective

Discuss whether unbiasedness is always the best property to prioritise. In practice, a biased estimator might have smaller mean squared error — what should scientists value more: unbiasedness or lower overall error?

🌐 EE focus

An EE might explore properties of estimators (biased vs unbiased) and compare MSE (mean squared error) for different estimators in simulations to justify estimator choice.

📌 Final checklist & common exam tasks

  • When given a linear transform, write both E(aX + b) and Var(aX + b) and compute numerically.
  • When summing independent variables use Var(sum) = sum Var — show independence assumption.
  • For sample statistics: show formula for x̄and s2n−1, state unbiasedness and compute sample standard error when asked.
  • If asked to interpret, always give a one-line plain-English sentence (e.g., “scaling by 3 triples the mean, variance multiplies by 9”).

🧠 Paper tip

  • Write the formula then substitute numbers. Examiners give method marks for clear symbolic steps even if arithmetic slips.
  • If a question mentions independence, explicitly include “assuming independence” when using Var(sum) = Σ Var.
  • Label units and give short interpretations of numerical answers.

❤️ CAS idea

Run a school survey; compute sample mean and sample variance for different classes, show how averaging reduces variance and explain the practical meaning.