the Pareto distribution
/ puh-RAY-toh /
A handful of people hold most of the wealth; a few cities dwarf all the rest; a tiny number of websites get the lion's share of traffic. These 'the big get enormous' patterns are the territory of the Pareto distribution, the classic heavy-tailed power law and the math behind the famous 80/20 rule.
Write X ~ Pareto(x_m, alpha) with a minimum value x_m > 0 and a shape parameter alpha > 0. Its tail is a power law: P(X > x) = (x_m / x)^alpha for x >= x_m. Crucially, this tail decays only polynomially — far slower than the exponential or normal — so very large values stay non-negligibly likely. The shape alpha controls how heavy the tail is and, alarmingly, which summaries even exist: the mean is finite only if alpha > 1 (and equals alpha x_m/(alpha - 1)), and the variance is finite only if alpha > 2. For small alpha the average is dominated by rare giant values and may not exist at all.
Pareto models wealth and income, city and firm sizes, file sizes, insurance losses, and earthquake magnitudes — anywhere a few cases are vastly larger than the typical one. The honest, important caveat: heavy tails break the comfortable intuitions built on the normal. The sample average can be unstable and misleading, 'typical' (the median or mode) is far below the mean, and a single extreme observation can dominate everything — so reasoning about a Pareto quantity by its average alone is a trap.
Suppose wealth follows Pareto with x_m = 10,000 and alpha = 1.5. Then P(wealth > 1,000,000) = (10,000 / 1,000,000)^1.5 = (0.01)^1.5 = 0.001 — one in a thousand, vastly more than a normal model would ever predict, and the variance does not even exist since alpha < 2.
A polynomial tail keeps giant values likely; small alpha can kill the mean entirely.
Heavy tails defy normal intuition: the mean exists only if alpha > 1 and the variance only if alpha > 2, so a sample average of Pareto data can be unstable or meaningless.