Concept

Extremal example

An object attaining a bound exactly, which is what turns a true inequality into the best one available. For a bound derived from stated moments the extremal object sits on as few points as there are constraints, and its support is where the proof's inequalities become equalities.

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

five values, unevenly weighted, and the mass outside 3 standard deviations. A distribution drawn as bars, with the windows one and a half, two and three standard deviations wide marked. The probability outside each window is summed and compared with the bound that knows only the variance.

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

The most mass 2 deviations out, with only a mean and a variance. The distribution putting as much probability as possible outside a window 2 standard deviations wide, found by searching every triple of support points, with the quadratic certificate that bounds it drawn over.

The bound is the answer to a search

Chebyshev's inequality is not a clever estimate that happens to be sharp. It is the exact answer to a maximisation over all distributions with a stated mean and variance, and the polynomial that proves nothing beats it is the certificate a search of that kind always produces.

probability · Concentration

Named alongside it

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

Concentration inequality Expectation Tail bound Variance Convergence rate Duality Exhaustive search Heavy tails Linear program Normal distribution Random walk Standard deviation

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