Expectation and variance

Definition

In simple words

A game pays ₹100 on heads and nothing on tails. You won't ever win exactly ₹50 in one toss, yet on average you win ₹50 a toss. That average is the expected value. A second game that always pays ₹50 has the same expected value but zero variance, because there's no risk.

Why it matters

Reinforcement learning maximises expected reward. Training minimises expected loss. The bias-variance trade-off, batch normalisation and weight initialisation all rely on variance. When a paper writes E[...], it means this.

How it works

• Expectation is linear: E[aX + bY] = aE[X] + bE[Y], and that holds even if X and Y are dependent. This trick shortens many proofs.
• Variance is not linear: Var(aX) = a²Var(X). Doubling a value multiplies its variance by four.
• For independent X and Y, Var(X + Y) = Var(X) + Var(Y).

The maths

For one roll of a fair die:

E[X]  = (1+2+3+4+5+6) / 6 = 3.5
E[X²] = (1+4+9+16+25+36) / 6 = 91/6 ≈ 15.17
Var   = 15.17 − 3.5² = 15.17 − 12.25 ≈ 2.92
σ     = √2.92 ≈ 1.71

In code

import numpy as np

faces = np.arange(1, 7)
probabilities = np.full(6, 1 / 6)

mean = np.sum(faces * probabilities)
variance = np.sum((faces - mean) ** 2 * probabilities)
print("Expected value:", round(mean, 2))
print("Variance:", round(variance, 2))
print("Standard deviation:", round(np.sqrt(variance), 2))

rng = np.random.default_rng(0)
rolls = rng.integers(1, 7, size=100_000)
print("Simulated mean:", round(rolls.mean(), 2), "variance:", round(rolls.var(), 2))

Expected value: 3.5
Variance: 2.92
Standard deviation: 1.71
Simulated mean: 3.5 variance: 2.91

Common mistakes

• Expecting the expected value to actually occur. A die never shows 3.5.
• Reading variance in the original units. It is in squared units, which is why standard deviation is easier to interpret.

Check yourself

Summary

• Expected value: the probability-weighted average.
• Variance: the average squared distance from the mean.
• Standard deviation: the square root of variance, in the original units.