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Concept

Heavy tails

A distribution whose extreme values are common enough to dominate averages, and whose variance or mean may not exist at all.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

1173760121153109683661left or right, 12 times, 600 times over

A bell curve assembled out of coin flips

Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.

probability · central limit -6-4-202460.000.100.200.300.400.50valueprobability1.5σ2σ3σthe meanmean 0.300, standard deviation 1.847; beyond 1.5 of them lies 0.2000 against a bound of 0.4444, beyond 2 ofthem lies 0.1000 against a bound of 0.2500, beyond 3 of them lies 0.0000 against a bound of 0.1111the bound knows only the variance — not the shape, not the number of values, not whether the distribution issymmetric — which is why it is so far from tight here and cannot be improved in general

How far from the average a thing can be

Knowing only an average and a spread — nothing about the shape, nothing about the number of outcomes, nothing about symmetry — the chance of landing three standard deviations out is at most one in nine. And there is a distribution that lands there exactly that often, so the bound cannot be improved.

probability · concentration

Named alongside it

The objects these essays reach for when they reach for this one.

Normal distributionBinomial distributionConcentration inequalityConvergenceConvergence rateExpectationExtremal exampleGalton boardIndependenceLimitPiRandom walk

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