moments
Moments are a family of numbers that summarize the shape of a distribution, one feature at a time. The first moment tells you where its center is, the second how wide it spreads, the third whether it leans left or right, the fourth how heavy its tails are. Together they build up an increasingly detailed portrait of a random variable from just a handful of averages.
Formally, the k-th raw moment is E[X^k] — the average of X raised to the power k. The first raw moment is the mean. More useful are the central moments, measured around the mean: the second central moment is the variance (spread); the third, scaled, gives skewness (asymmetry — a long right tail means positive skew); the fourth, scaled, gives kurtosis (how much weight sits in the tails versus the middle). So the mean, variance, skewness, and kurtosis are just the first four moments dressed up in plain language.
Moments are how actuaries describe and compare loss distributions without drawing every curve. The method of moments fits a distribution by matching its theoretical moments to the moments seen in the data. Skewness matters intensely in insurance because losses are almost always right-skewed — many small claims, a few enormous ones — and a model that captures only mean and variance will badly understate the chance of a catastrophic claim. High kurtosis is the fingerprint of the fat tails that drive capital requirements.
A fire-loss distribution might have mean $3,000 and a strong positive skew: most claims are small, but the rare warehouse fire pulls the third moment up. Two distributions with the same mean and variance can still differ sharply in skew, and so in their tail risk.
Skewness — the third moment — is what warns you about rare giant losses.
Some heavy-tailed distributions actuaries use (certain Paretos) have infinite variance or even infinite mean — their higher moments simply do not exist, so methods that assume all moments are finite quietly fail on exactly the riskiest data.