Expectation and variance
Objective. Calculate moments and use linearity of expectation.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
The expectation E(X) of a random variable is its probability-weighted average value, calculated as Σ xp(x) for discrete X, representing the long-run mean over many repeated trials. Variance Var(X) = E(X²) − [E(X)]² measures the spread of X around this mean, with larger variance indicating more variability in outcomes. Linearity of expectation states E(aX + b) = aE(X) + b and crucially E(X + Y) = E(X) + E(Y) even when X and Y are dependent, making it a remarkably powerful computational shortcut. For a fair die, E(X) = 3.5 and E(X²) = 91/6, so Var(X) = 91/6 − 3.5² = 35/12.
Worked example
A random variable X has P(X=1) = 0.3, P(X=2) = 0.5, P(X=3) = 0.2. Find the expectation E(X).
- 1Recall E(X) = Σ xp(x), multiplying each value by its probability.
- 2Compute each term: 1(0.3) = 0.3, 2(0.5) = 1.0, 3(0.2) = 0.6.
- 3Add these weighted values together: 0.3 + 1.0 + 0.6.
- 4Simplify the sum to obtain the expectation.
Answer: E(X) = 1.9
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Start practisingCommon mistake. Students assume Var(aX) = aVar(X), forgetting the scaling factor is squared; since variance measures squared deviations, the correct identity is Var(aX) = a²Var(X).
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