the central limit theorem

The central limit theorem is one of the most surprising and useful facts in all of probability: add up or average a large number of independent random pieces, and the result looks like a normal bell curve — no matter what shape the individual pieces had. The pieces can be skewed, lumpy, discrete, ugly; their sum, once there are enough of them, smooths into the familiar symmetric bell. It is why the normal distribution turns up everywhere.

Precisely, if you sum many independent random variables that each have a finite mean and variance, the distribution of that sum (or of their average) approaches a normal distribution as the number grows. The sum's mean is the sum of the individual means, and its variance is the sum of the individual variances (when independent). So even though no single policy's loss is normal, the total loss of a large book tends toward normality, with a mean and spread you can compute by simple addition. A handful of terms is not enough; the approximation gets better the more independent pieces you pile on, and slower when the pieces are very skewed.

For actuaries the central limit theorem is the bridge from messy individual risks to tractable portfolio totals. It justifies approximating aggregate claims, the sum of many small losses, as normal — which underlies practical capital calculations, the normal-approximation to the binomial and Poisson, and many confidence-interval and ruin formulas. But its caveats are vital: it describes the centre of the distribution far better than the tails, and it needs finite variance and genuine independence. For heavy-tailed losses (Pareto-like) or correlated catastrophe risks, the normal approximation can dangerously understate exactly the extreme outcomes an insurer most needs to get right.