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

Find E(X), then E(X²) by squaring each value before weighting, and subtract: Var(X) = E(X²) − [E(X)]².

Does DSE M1 include sums of random variables?

No. The curriculum notes state that sums and differences of random variables and joint probability distributions are not required.

What does the expectation mean?

It is the long-run average value of X over many repetitions of the experiment. It need not be a value X can actually take: here E(X) = 1.7.

Discrete Random Variables, Expectation and Variance (DSE M1)

A discrete random variable X takes separate values, each with a probability, and the probabilities add up to 1. Its expectation is E(X) = Σ x P(X = x) and its variance is Var(X) = E(X²) − [E(X)]².

Solve Discrete Random Variables: Expectation and Variance questions with AI

The distribution of Example 1 as a bar chart, with the expectation E(X) = 1.7 marked by the dashed line. Try it yourself →

Expectation and Variance Formulas

Probabilities0 ≤ P(X = x) ≤ 1, Σ P(X = x) = 1

ExpectationE(X) = Σ x P(X = x)

Function of XE[g(X)] = Σ g(x) P(X = x)

VarianceVar(X) = E[(X − μ)²] = E(X²) − [E(X)]²

Linear changeE(aX + b) = aE(X) + b, Var(aX + b) = a²Var(X)

Worked Examples — Discrete Random Variables: Expectation and Variance

Example 1: Expectation and variance from a table

X takes the values 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2. Find E(X), Var(X), E(3X + 2) and Var(3X + 2).

E(X) = 0(0.1) + 1(0.3) + 2(0.4) + 3(0.2) = 1.7
E(X²) = 0(0.1) + 1(0.3) + 4(0.4) + 9(0.2) = 3.7
Var(X) = 3.7 − 1.7² = 0.81
E(3X + 2) = 3(1.7) + 2 = 7.1
Var(3X + 2) = 3² × 0.81 = 7.29

Answer: E(X) = 1.7, Var(X) = 0.81, E(3X + 2) = 7.1, Var(3X + 2) = 7.29

Example 2: Finding an unknown constant in a distribution

The probability function of X is P(X = x) = kx for x = 1, 2, 3, 4. Find k and E(X).

The probabilities add up to 1: k(1 + 2 + 3 + 4) = 1
k = 0.1
E(X) = 0.1(1² + 2² + 3² + 4²) = 0.1 × 30 = 3

Answer: k = 0.1, E(X) = 3

💡 Exam Traps & Secrets

⚠️ Var(aX + b) uses a², and b disappearsAdding a constant shifts every value by the same amount, so the spread is unchanged. Multiplying by a multiplies the variance by a², never by a.

⚠️ E(X²) is not [E(X)]²Square the values first, then weight by the probabilities. In Example 1, E(X²) = 3.7 while [E(X)]² = 2.89; the difference is the variance.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(E(X) and Var(X))X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3. Find E(X) and Var(X) and show the distribution as a bar chart.

(Linear change)If E(X) = 4 and Var(X) = 2.5, find E(2X − 3) and Var(2X − 3).

(Uniform distribution)A fair die is thrown. Let X be the score. Find E(X) and Var(X).