Concept

Binomial distribution

The distribution of how many successes come out of a fixed number of independent trials with the same chance each. Its mean is the number of trials times the chance of success, and its spread grows like the square root of that number.

Named by 4 essays across 3 fields — 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

Nine walks, and the square root. 9 independent walks of 400 steps, each step one place left or right. The dashed curves are ±√n: the walks stay near them, spill past them, and come back — which is what a typical distance means as opposed to a limit.

A walk that always comes home, until it does not

Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.

probability · Random walk

A polygon of 4 points averaged down to its curve at t = 0.40. A control polygon of 4 points, the 3 rounds of weighted averaging at t = 0.40 drawn as nested polylines, the single point they end at, and the whole curve those points trace. The weights on the control points are 0.216, 0.432, 0.288, 0.064.

Averaging down the triangle

Change one word in the rule that builds Pascal's triangle — take a share of each entry above instead of adding them — and the triangle stops counting and starts averaging. The same rule then draws smooth curves from polygons and approximates every continuous function by polynomials, at a rate that no amount of smoothness can improve.

discrete · Pascals triangle

Channel capacity 1 − H(p), and three codes at p = 0.1. The capacity of the binary symmetric channel plotted against its flip probability, with the rates of repetition, the Hamming code and no coding marked at one flip probability.

The rate a noisy channel allows

A channel that flips one bit in ten can still carry messages with as few errors as anyone likes — at up to 0.531 message bits per transmitted bit, and at no rate above that. The number is Shannon's capacity, 1 − H(p). Repetition reaches reliability only by sending nothing; a code chosen at random gets there at any rate below the limit; and the reason there is a limit at all is a count of how many flip patterns a block of noise can hold.

computation · Error-correcting codes

Named alongside it

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

Convergence Independence Limit Normal distribution Approximation Binomial coefficient Channel capacity Convex hull Diffusion Entropy Error-correcting code Galton board

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