Combinations of Random Variables

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Edexcel Further Stats 2 · Chapter 4

Combinations of Random Variables

Big idea: if you add or subtract normal random variables (each multiplied by a number), the result is STILL normal — you just add up the new means and, separately, add up the new variances (never standard deviations directly).

Summary — What This Chapter Is Really About

You already know E(X+Y) = E(X)+E(Y) and similar rules for expectation — this chapter reuses them.
For independent variables, variances always add — even when you're subtracting the variables themselves.
If X and Y are each normally distributed and independent, then aX + bY (or aX − bY) is also normally distributed.
The new normal distribution's mean and variance are found using simple formulas — then you standardise with Z exactly like any other normal distribution question.
When you add several copies of the same distribution (like 7 identical bags of sugar), there's a shortcut: just multiply the mean and variance by n.
The single biggest exam trap: mixing up variance and standard deviation when combining. This chapter is built almost entirely around avoiding that one mistake.

1. Combining Random Variables — The Expectation Rules

Let's start with something you've seen before, just to warm up. If X and Y are any two random variables (they don't even need to be independent, and they don't even need to be normal), then expectation behaves exactly like you'd hope — it's "linear," meaning it distributes nicely over addition, subtraction, and multiplication by constants.

Expectation rules

E(X + Y) = E(X) + E(Y)

E(X − Y) = E(X) − E(Y)

E(aX + bY) = aE(X) + bE(Y)

E(aX − bY) = aE(X) − bE(Y)

In plain words: to find the average of a combination of variables, just combine the averages the same way. Multiplying a variable by a constant "a" scales its mean by a too.

Why this feels obvious If the average apple weighs 125g and the average orange weighs 250g, then obviously the average weight of "3 apples + 3 oranges" is 3(125) + 3(250). You're not doing anything clever here — you're just scaling and adding averages the same way you'd scale and add the actual quantities.

2. Combining Random Variables — The Variance Rules (the important one)

This is where people trip up, so slow down here. Variance rules only work this simply when X and Y are independent — meaning the outcome of one doesn't affect or relate to the outcome of the other (e.g. the mass of an apple and the mass of an orange in the same bag are independent; your maths score and your physics score on the same day probably aren't).

Variance rules — for independent X and Y only

Var(X + Y) = Var(X) + Var(Y)

Var(X − Y) = Var(X) + Var(Y)

Var(aX + bY) = a²Var(X) + b²Var(Y)

Var(aX − bY) = a²Var(X) + b²Var(Y)

⚠️ The #1 exam mistake Notice that Var(X − Y) = Var(X) + Var(Y) — the minus sign disappears and becomes a plus! This feels wrong the first time you see it, but think about it intuitively: subtracting two independent random things doesn't cancel out uncertainty, it adds MORE uncertainty (now you have two separate sources of randomness contributing to the spread of the result). Variance never subtracts when the variables are independent.

Why does "a" become "a²"?

3. Combining Two Normal Distributions

also

Core formula — memorise this exactly

aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²)

How to read N(μ, σ²)

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Also in the full note

4. The "Difference" Trick — Turning P(3X > Y) into a Single Distribution
5. Combining Several Copies of the SAME Distribution
What to Memorise
Concepts Checklist
Exam Tips — Where Marks Are Lost
Worked Example — Full Method
Worked Example

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Combinations of Random Variables