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Concept

Probability density

A non-negative function whose integral over a region gives the chance of landing in it, and whose integral over everything is one. It is the continuous counterpart of a list of probabilities, and its value at a point is a rate rather than a chance.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Two unbiased estimates of one integral, and their spread. The sharply peaked integrand with the proposal density that follows it, above a strip plot of 200 estimates from each of two methods; the weighted estimates cluster 4.2 times more tightly about the same value.

Sampling where the answer lives

Monte Carlo error cannot be made to fall faster than the square root, so the only thing left to attack is the constant in front of it. Drawing points where the integrand is large, and dividing by how often they were drawn, leaves the answer alone and can shrink the noise many times over.

probability · Monte Carlo A walk that visits each state as often as its weight says. The target distribution over 12 states with the share of a 40,000-step Metropolis run beside each bar, above the first 300 steps of the walk itself; the two distributions differ by 0.5 per cent in total.

A walk that samples a distribution

When a distribution can be evaluated but not drawn from, a wandering point can be arranged to visit each state as often as its weight says. The rule needs no normalising constant, compares two weights and steps or stays.

probability · Monte Carlo 12 harmonic series with random signs, each settling on its own sum. Partial sums of the harmonic series with each sign chosen by a fair coin, for several independent runs, plotted against the number of terms on a logarithmic scale, beside the all-plus and alternating sign patterns.

A coin in front of every term

Put all plus signs in front of 1, 1/2, 1/3, … and the sum runs off to infinity; alternate them and it settles on log 2. Toss a fair coin for each sign instead, and the sum settles — every time, on a different number. Where it tends to settle has a smooth, flat-topped shape, and at the value 2 that shape takes a height that agrees with one eighth to forty-two decimal places and is not one eighth.

analysis · Harmonic series Where a coin-signed geometric series lands, for λ = 1/3, 1/2, 1/√2, 1/φ. Histograms of the exact distribution of the sum of plus or minus λ to the k, one panel per value of λ: a dust of separated pieces below one half, a flat block at one half, and smooth-looking overlapping shapes above.

A coin in front of every power

Toss a coin for each sign of ±1 ± λ ± λ² ± … and the sum lands somewhere. Below λ = 1/2 it lands on a dust with gaps in it, at exactly 1/2 it lands anywhere with equal chance, and above 1/2 it lands on a smooth-looking hill — which for most λ has a density and for the golden value does not, though no picture can tell the two apart.

analysis · Harmonic series Sign patterns, and the far fewer points they land on. A logarithmic plot of the number of sign patterns, two to the n, against the number of distinct values the golden geometric sum takes, which is a Fibonacci number less one and falls further behind at every step.

Two sign patterns that land together

At λ = 1/φ the sign patterns + − − and − + + land in exactly the same place, because λ² + λ = 1. That one coincidence, repeated wherever it fits, puts 2ⁿ patterns onto a Fibonacci number of points, leaves the random sum's transform ringing at the same height forever, and makes a distribution that fills a whole interval live on a set of no length.

analysis · Harmonic series Bands of a sphere and of its cylinder, cut by the same planes. A unit sphere inside its cylinder with 3 horizontal bands; each sphere band has area 1.885, 1.885, 1.885, equal to the cylinder band of the same height.

A band of a sphere is a band of its cylinder

Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.

geometry · Circle area Products of widening sinc waves, all with area π/2 but the last. Graphs of the products of sinc(x/(2k+1)) for k = 0 to n, n = 0, 1, 3, 7, on 0 ≤ x ≤ 24.

Seven times half of π, and then not

The area under sin x / x, out to infinity, is π/2. Multiply in sin(x/3)/(x/3) and the area is still π/2; multiply in the waves for 5, 7, 9, 11 and 13 and it is still π/2, exactly. Add the wave for 15 and the area falls short — by about 2.3 × 10⁻¹¹. Nothing about the curves changes at the eighth factor. What changes is that 1/3 + 1/5 + … + 1/15 has just passed 1.

analysis · Circular functions

Named alongside it

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

Almost surelyBernoulli convolutionBinary expansionCharacteristic functionFractal dimensionHarmonic seriesIntegralMonte CarloPiSamplingVarianceAlgebraic integer

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