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Concept

Expectation

The average of a quantity's values, each weighted by how likely it is.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

10 → 11.201 → 21.502 → 323 → 434 → 565 → 614.70 draws expected in all6 kinds, and the last is the expensive onethe waits are 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.70 drawsthe last one alone costs 6 draws on average, which is why the total grows faster than the number of kinds

How long until every one turns up

Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.

probability · expectation -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 -1-0.50.510.01.02.03.0the average, minus the meann = 4n = 16n = 64-3-2-11230.00.10.20.30.4the sum, minus n means, over √nn = 4n = 16n = 64the same exact distributions, drawn twice as densities: divided by n the spread falls 0.500 → 0.250 →0.125, divided by √n it is 1.000 every timeso the left curves climb — 0.75 → 1.57 → 3.18 at the peak — and the right ones settle on 0.399, which is theheight of the bell curve

The average settles and the wobble does not

Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.

probability · central limit 01234560.000.100.200.300.400.50how many are looked at and passed overchance of taking the best14.3%35.0%41.4%40.7%35.2%26.2%14.3%every one of the 5,040 orders, for every threshold: looking at 2 of 7 and then taking the first that beats them wins2,088 times, which is 41.4%taking the first one wins 14.3% and so does taking the last, because both amount to choosing without looking —the rule beats them by a factor of 2.90

When to stop looking

Candidates arrive one at a time in a random order. Each must be accepted or rejected on the spot, with no going back and no way to know what is still to come. The best possible rule is to look at about a third of them and then take the first one that beats everything seen — and it works about a third of the time, however many there are.

probability · optimal stopping

Named alongside it

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

Convergence rateHarmonic seriesNormal distributionVarianceApproximationCentral limit theoremCollisionConcentration inequalityConditional probabilityConvergenceCounting argumentDecision procedure

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