Expectation, Mean, and Variance

A PMF is a complete description of a random variable, but a full distribution is unwieldy to quote in a single sentence — a trader wants "the expected P&L" and "how risky is it," not the whole payoff distribution table. Expectation ($\mathbb{E}[X]$) is the probability-weighted average value; variance ($\text{Var}(X)$) is the average squared deviation from that average, the canonical measure of spread. This note defines both directly from the PMF, develops the mean's physical "centre of gravity" interpretation, and works a full example culminating in the observation — deliberately left as friction here — that variance computed straight from the definition requires the PMF of a transformed random variable, a tedium resolved in the next lesson's expected value rule.

Learning objectives. By the end of this note you should be able to: