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Concept

Uniform distribution

The distribution that gives every outcome in a set the same probability, or every interval of the same length the same chance. It is the model of a fair choice, and checking that a procedure really produces it is often harder than it looks.

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

Points too even to be random. 256 independent random points beside 256 points of a Halton sequence, with the largest mismatch between a box's share of points and its area plotted against the number of points for both.

Points too even to be random

Independent random points clump, and the clumping is what makes the error fall only as the square root. Points chosen to be evenly spread rather than independently beat that rate, and the price is that nothing about them is random at all.

probability · Monte Carlo Fifty-two cards, swapped the naive way. Naive shuffle, n=52: max ratio 1.3522 at card 2 → place 1; min ratio 0.7429.

A perfect coin and a biased shuffle

Give a program a perfect source of random numbers and ask it to shuffle a deck by swapping each card with a card chosen from anywhere in the deck. The result is biased, provably and for every deck of three or more cards: there are nⁿ equally likely runs and n! orders, and the first number is never a multiple of the second. In a deck of 52 some card-and-place pairs come up a third more often than they should. Swapping only with cards not yet placed fixes it; swapping only with later cards makes nothing but single cycles.

computation · Pseudorandomness

Named alongside it

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

Convergence rateCycleDiscrepancyDivisibilityEquidistributionError boundIntegralMonte CarloPermutationPseudorandomnessQuasi-random sequenceRandom permutation

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