expectation
The expectation of a random variable is its long-run average — the number you would converge to if you could repeat the experiment over and over and average the results. It is the single best one-number summary of where a random quantity tends to sit, the balance point of its distribution. We write it E[X] and often call it the mean.
You compute it as a probability-weighted average of the possible values: multiply each value by its probability and add them all up (for a continuous variable, integrate value times density). If a lottery pays $0 with probability 0.99 and $100 with probability 0.01, its expectation is 0×0.99 + 100×0.01 = $1, even though you can never actually win exactly $1. A wonderfully useful property is linearity: E[aX + bY] = a·E[X] + b·E[Y] holds always, even when X and Y are dependent — which is why expectations of sums are so easy to handle.
Expectation is the foundation stone of insurance pricing. The pure premium — the part of the price meant to cover claims — is essentially E[loss], the expected cost of the risk transferred. The actuarial present value of a benefit is an expectation. The equivalence principle sets premiums so that the expected present value of premiums equals the expected present value of benefits. But a crucial caution: the expectation alone says nothing about risk; two policies can share the same E[loss] while one is placid and the other capable of ruinous swings.
A policy has a 2% chance of a $50,000 claim and otherwise pays nothing. E[loss] = 0.02 × 50,000 = $1,000. That $1,000 is the pure premium — before expenses, profit, and a margin for risk.
The pure premium is just the expected loss; risk margins come on top.
The expectation can be a value the variable never actually takes (like the '$1' lottery), and it tells you nothing about volatility — pricing on E[X] alone, with no margin for spread, would eventually bankrupt an insurer.