6.2 Linear Combinations of Random Variables

2026 Syllabus Objectives

By the end of this topic, you should be able to use the following results when solving problems:

  1. E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X)
  2. E(aX + bY) = aE(X) + bE(Y)
  3. Var(aX + bY) = a²Var(X) + b²Var(Y) — but only when X and Y are independent
  4. If X has a normal distribution, then aX + b also has a normal distribution
  5. If X and Y have independent normal distributions, then aX + bY has a normal distribution
  6. If X and Y have independent Poisson distributions, then X + Y has a Poisson distribution

Note: You do not need to prove any of these results. You just need to be able to use them correctly.

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Part 1 — Quick Recap: Expectation and Variance

Before diving into linear combinations, let's briefly recall what expectation (mean) and variance are.

  • The expectation of a random variable X, written E(X), is the long-run average value you would expect if you repeated the experiment many times.
  • The variance of X, written Var(X), measures how spread out the values of X are around the mean.

These are the two key numbers that describe any random variable. Everything in this topic builds on them.

Part 2 — Scaling and Shifting a Single Random Variable

What does aX + b mean?

Imagine you have a random variable X. Now suppose you:

  • Multiply every value of X by a constant a (this is called scaling)
  • Add a constant b to every value (this is called shifting)

The result is a new random variable: aX + b.

For example, if X is the temperature in Celsius, then 1.8X + 32 converts it to Fahrenheit.

How does E(aX + b) work?

When you scale X by a and shift by b, the mean also scales and shifts in the same way:

E(aX+b)=aE(X)+b\boxed{E(aX + b) = aE(X) + b}

In plain English: Multiply the mean by a, then add b.

Example: Suppose E(X) = 5. Then:

  • E(3X + 2) = 3 × 5 + 2 = 17
  • E(10X − 4) = 10 × 5 − 4 = 46

How does Var(aX + b) work?

When you scale and shift, the variance changes differently:

Var(aX+b)=a2⋅Var(X)\boxed{Var(aX + b) = a^2 \cdot Var(X)}

Two key points to notice:

  1. The constant b disappears. Adding or subtracting a fixed number shifts all the values by the same amount, so the spread doesn't change — only the centre does. This means adding b has no effect on variance.
  2. The constant a gets squared. When you multiply all values by a, the distances between them also multiply by a, so the variance multiplies by a².

Example: Suppose Var(X) = 4. Then:

  • Var(3X + 2) = 3² × 4 = 9 × 4 = 36
  • Var(3X − 7) = 3² × 4 = 36 (the −7 makes no difference)
  • Var(−2X + 1) = (−2)² × 4 = 4 × 4 = 16

Common mistake: Students sometimes write Var(aX + b) = a²Var(X) + b. Remember — the b vanishes completely. The +b only affects the mean, not the variance.

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