Matpic
mathematics, in pictures
Concept

Heavy tails

A distribution whose extreme values are common enough to dominate averages, and whose variance or mean may not exist at all. The largest of a sample is then comparable to the whole sum, which is why averaging such quantities buys so little.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

A Galton board after 600 balls. 600 balls fall through 12 rows of pegs, each bouncing left or right at random, and pile up in a bell-shaped heap.

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 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 Averages of a heavy-tailed quantity, which never settle. Running averages of draws from a Cauchy distribution, which jump rather than converge, beside the cumulative distributions of averages of 1, 4 and 16 draws, which lie on top of one another.

An average that never settles

The average of many independent quantities is supposed to steady as their number grows. For one famous distribution it does not steady at all — the average of a thousand draws has exactly the same distribution as a single draw, and no amount of further averaging changes it.

probability · Central limit How fast a sum becomes a bell curve. The largest gap between the distribution of a standardised sum and the bell curve, against the number of terms, on logarithmic axes. Both summands fall along a line of slope about minus a half.

How fast the bell arrives

The limit theorem says a standardised sum approaches the bell curve and says nothing about when. The rate is one over the square root of the number of terms, the constant in front is made of the third moment, and both are visible.

probability · Central limit The chance the average clears 0.75, against the number of draws. The exact probability that the average of n draws exceeds a fixed level, on a logarithmic scale, falling along a straight line whose slope is the rate function, with the normal approximation drawn beside it and diverging.

The tail is not a bell

The limit theorem describes a window of width one over the root of n around the mean; ask instead for the chance that an average lands a fixed distance away and the answer falls exponentially, at a rate computed from the summand before any n is chosen.

probability · Central limit

Named alongside it

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

Normal distributionExpectationVarianceConcentration inequalityConvergenceConvergence rateCounterexampleIndependenceScalingTail boundApproximationBinomial distribution

All concepts