So far we have asked whether events happen. Often we care about a number attached to a random outcome: a payoff, a count, a measurement. A random variable is that number, and two summaries, expectation and variance, capture its behaviour.
Random variables
A random variable assigns a number to each outcome of a random process. The result of a die roll is a random variable taking values to ; the number of heads in ten tosses is a random variable from to ; a gambler's winnings on a bet is one too. Each possible value comes with a probability, and together those form the variable's distribution.
Expected value
The expected value is the long-run average: what you would get per trial, averaged over very many trials. You compute it by weighting each value by its probability and adding:
For a fair die, . No single roll gives , but the average of many rolls homes in on it. Expected value is the backbone of decision-making under risk: a bet is favourable exactly when its expected payoff is positive, which is why, over time, the house always wins at games with negative expectation.
Variance
Expectation tells you the centre; it says nothing about the spread. Two bets can share an expected value while one is steady and the other wild. Variance measures that spread (the average squared distance of the outcomes from the mean) and its square root, the standard deviation, puts the spread back in the original units. A small variance means outcomes huddle near the average; a large one means they scatter widely.
Why they matter
Between them, expectation and variance summarise a random quantity in two numbers: where it tends to sit and how much it wobbles. That is often all a decision needs: an insurer pricing a policy, an investor weighing return against risk, a scientist reporting a measurement with its uncertainty. And these two numbers are exactly the handles on the distributions we meet next, above all the bell curve, which is defined by its mean and its spread.