What is a power-law distribution and how do you spot one?

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Answer

$P(X > x) \propto x^(-α) for large x.$
$Very heavy right tail — a few extreme observations dominate the mean.$
$Examples: word frequencies, city sizes, wealth, network degree distributions (Zipf's / Pareto's law).$
$\operatorname{log}(\operatorname{rank})$
$Consequences: sample means are unstable, need heavy-tailed methods (median, quantile regression, robust stats).$

Check yourself — multiple choice

Very light tail
P(X>x) ∝ x^(-α); heavy right tail; identified via log-log plot; median/robust stats needed
Symmetric
Same as normal

Power law: heavy tail; log-log linear; needs robust stats.

#distributions

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