HL AA · Statistics and Probability

Discrete probability distributions — expectation and variance (IB Math AA HL)

Updated October 2026

Quick answer: For a discrete random variable, the probabilities add to 1, E(X) = Σ x P(X = x) and Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x² P(X = x). For a linear change, E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).

E(X), Var(X), and the shortcut Var(X) = E(X²) - [E(X)]². HL AA Paper 1 tests these algebraically without a GDC.

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Key ideas

Use ΣP(X = x) = 1 to find an unknown constant k in the distribution.
E(X) is the long-run average, not necessarily a value X can take.
Var(X) = E(X²) − [E(X)]²: square the values (not the probabilities) for E(X²).
Adding a constant does not change the variance; multiplying by a multiplies the variance by a².
A fair game has expected gain 0. Set E(gain) = 0 to find a fair price or prize.

Worked example

The random variable X has P(X = x) = kx for x = 1, 2, 3, 4. Find k, E(X) and Var(X), and hence E(2X + 3) and Var(2X + 3).

k(1 + 2 + 3 + 4) = 1, so 10k = 1 and k = 1/10.
E(X) = (1 × 1 + 2 × 2 + 3 × 3 + 4 × 4) ÷ 10 = 30 ÷ 10 = 3.
E(X²) = (1 × 1 + 4 × 2 + 9 × 3 + 16 × 4) ÷ 10 = 100 ÷ 10 = 10.
Var(X) = 10 − 3² = 1.
E(2X + 3) = 2 × 3 + 3 = 9 and Var(2X + 3) = 2² × 1 = 4.

Answer: k = 1/10, E(X) = 3, Var(X) = 1, E(2X + 3) = 9, Var(2X + 3) = 4.

Exam-style HL AA questions on this topic

Advanced Random Variables questions — 52 questions with mark schemes, plus free worked examples
Conditional Probability questions — 33 questions with mark schemes, plus free worked examples
All HL AA Statistics and Probability questions by topic

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5 quick questions on discrete random variables. No sign-in needed.

Find \(E(X)\) for the distribution in the table.

\(x\)	10	20	50
\(P(X = x)\)	\(\frac{7}{10}\)	\(\frac{1}{5}\)	\(\frac{1}{10}\)

Choose your answer · keys 1–4 \(400\)\(\frac{80}{3}\)\(1\)\(16\)

Find \(E(X)\) for the distribution in the table.

\(x\)	1	2	3	4
\(P(X = x)\)	\(\frac{1}{10}\)	\(\frac{1}{5}\)	\(\frac{3}{10}\)	\(\frac{2}{5}\)

Choose your answer · keys 1–4 \(1\)\(10\)\(3\)\(2.5\)

Find \(E(X)\) for the distribution in the table.

\(x\)	0	1	2
\(P(X = x)\)	\(\frac{1}{4}\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)

Choose your answer · keys 1–4 \(1\)\(\frac{1}{3}\)\(3\)\(1.5\)

Find \(E(X)\) for the distribution in the table.

\(x\)	2	5	10
\(P(X = x)\)	\(\frac{1}{2}\)	\(\frac{3}{10}\)	\(\frac{1}{5}\)

Choose your answer · keys 1–4 \(\frac{17}{3}\)\(29.5\)\(4.5\)\(1\)

Find \(E(X)\) for the distribution in the table.

\(x\)	-1	0	3
\(P(X = x)\)	\(\frac{2}{5}\)	\(\frac{2}{5}\)	\(\frac{1}{5}\)

Choose your answer · keys 1–4 \(\frac{2}{3}\)\(1\)\(2.2\)\(0.2\)

Maths IA ideas that use discrete probability distributions

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Revise Discrete probability distributions for the exam

HL AA revision notes — every unit summarised, with the key results to learn
HL AA past papers — specimen papers and timed practice papers
Formulas for Statistics and Probability — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original probability problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S33 · Problem S106 · Problem S34 · Problem S107. All 20 →

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FAQ

How do you find the variance of a discrete random variable?

Find E(X) = Σ x P(X = x) and E(X²) = Σ x² P(X = x), then Var(X) = E(X²) − [E(X)]². Check that the variance is not negative: if it is, there is an arithmetic slip.

Where can I practise discrete probability distributions questions?

On the Advanced Random Variables questions page: 52 exam-style HL AA questions with full mark schemes, plus free step-by-step worked examples.

Can I use my calculator for discrete probability distributions?

In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one.

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