Average over possibilities.

A discrete random variable assigns a number to each outcome. Every probability must lie between zero and one, and the probabilities must sum to one. Expected value is a probability-weighted average, which need not be a possible individual outcome.

Variance measures weighted squared distance from the mean. The equivalent calculation E( $X^{2}$ ) − [E(X)]² is often convenient. Standard deviation is the square root of variance and has the same units as X.

INTERACTIVE MODEL

Try it. Watch it change.

P(X = 0)0.4 P(X = 1)0.3333

P(X = 1) = 0.3333 · P(X ≤ 1) = 0.7333
Mean = 0.8667 · variance = 0.6489

Select a bar to read its exact model probability. The probabilities sum to one; bars represent discrete outcomes.

Explore: Redistribute probability among 0, 1 and 2. Keep the total at one and compare the mean and spread.

Expected value

E(X)=\sum_x xp(x)

Variance

\operatorname{Var}(X)=E(X^2)-[E(X)]^2

WORKED EXAMPLE

X is 0 or 4 with equal probabilities. Find its mean and variance.

E(X) = 0(0.5) + 4(0.5) = 2.
E( $X^{2}$ ) = 0 + 16(0.5) = 8.
Var(X) = 8 − $2^{2}$ = 4.

Assumed knowledge

Weighted averages and squared units.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Validate a probability distribution.
Calculate expectation, variance and standard deviation.

QCAA Mathematical Methods 2025 v1.3 · p. 26 ↗

READY TO TRY IT?

Average over possibilities.

X is 0 or 4 with equal probabilities. Find its mean and variance.

Put the idea to work.