expectation (the expected value)

Imagine you play a game many, many times and write down your payoff each time. If you average all those numbers, where does the average settle? That long-run average is the expectation. It is the single number that summarizes the 'centre of gravity' of a random variable — the value you should plan around if you face the same situation over and over.

For a discrete random variable X that takes value x with probability P(X = x), the expectation is a probability-weighted average: E[X] = sum over all x of x times P(X = x). You walk through every possible value, multiply it by how likely it is, and add up. For a fair die, E[X] = 1(1/6) + 2(1/6) + ... + 6(1/6) = 3.5. Notice 3.5 is not even a face you can roll — the expectation need not be a possible outcome. It is a balance point, the spot where the distribution would balance if each probability were a weight on a seesaw.

Expectation is the workhorse of probability: it feeds variance, covariance, moments, and almost every limit theorem. But it can mislead. For a lottery the expectation is tiny and negative, yet people play for the rare huge payoff. And for some heavy-tailed distributions (the Cauchy is the famous example) the sum that defines E[X] does not even converge, so the mean simply does not exist. Expectation is one summary, not the whole story.