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Concept

Distribution

A description of how a quantity's values are spread: which values occur, and how often or with what probability. It can be given by a histogram, a density or a cumulative function, and two quantities with the same distribution cannot be told apart by any statistic.

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

Time spent on one side of the axis. The exact distribution of the number of steps a 40-step fair walk spends above the axis. It is U-shaped: the extremes are the likeliest outcomes and an even split is the rarest.

Half the time is the rarest answer

In a fair game of many rounds, the fraction of the time one side is ahead is not usually near a half. It is usually near nought or one, and an even split is the single least likely outcome there is.

probability · Random walk How the abundancy σ(n)/n is distributed, up to 1,000,000. The share of numbers up to 1000000 whose ratio σ(n)/n is at least t, for t from 1 to 4. It is 0.2475 at two, 0.0202 at three and 0.00023 at four, with a steep fall just above one, where the numbers with no small prime factor sit.

Why a quarter of numbers overshoot

About one number in four has proper divisors adding to more than itself. That a proportion exists at all is not automatic — there are sets defined just as simply that have no proportion — and the reason this one does is that abundance is inherited by multiples, and the numbers it is first inherited from are sparse enough to add up.

number · Perfect numbers

Named alongside it

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

AbundanceArcsine lawCentral limitConvergenceDensityDivisor sumFairnessFirst returnMultiplicative functionPerfect numberPrimesRandom walk

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