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mathematics, in pictures
The collection

Every essay — page 23

Page 23 of 47, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Probability

Randomness with a shape.

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

7 figures 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

7 figures The 91 histograms 12 draws can produce. A triangle whose points are the possible histograms of a fixed number of draws over three faces, each drawn as a dot shaded by how far it is from the true distribution.

When the whole histogram deviates

7 figures The chance of being connected, against the chance of an edge. Curves of the exact probability that a random graph on three to six labelled points is connected, plotted against the probability of each individual edge.

The moment everything joins up

6 figures How fast each way of averaging closes in, as the dimension grows. Relative error against the number of points, both on logarithmic scales, for a regular grid in 1, 4, 8 dimensions and for random points in 8; the grid's lines steepen or flatten with the dimension and the random one does not move from a slope of a half.

The error that does not care how many dimensions

6 figures 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

6 figures 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

7 figures 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

6 figures A game that stops, over totals 0 to 5. States in a row with arrows up and down between them and the two ends absorbing, above a table of the expected number of steps and the chance of ending at the top from each start.

The chain that stops

6 figures A walk on a weighted graph, and a cycle whose traffic goes one way. A weighted graph with the long-run share of each state read off its total weight, beside a three-state cycle whose shares are equal and whose traffic circulates.

The chain that runs the same backwards

6 figures The time a single walk spends in each state, against the share it should hold. Paired bars for each state, one the fraction of a long run's time spent there and one the computed stationary share, above a table of expected return times.

The time spent and the share held

6 figures One walk on the whole numbers, three chances, three different fates. The relative weight of each state for three step-up chances, drawn as bars, with the running total of those weights and what each case means beneath.

Where the shares have nowhere to go

7 figures How fast a chain forgets where it started. The total variation distance to the stationary distribution plotted logarithmically against the number of steps, for each of 3 starting states. The curves are straight lines of equal slope.

How long until it forgets

5 figures The moment a giant piece appears. The largest component's share of 900 points plotted against the average degree, with the measured values as dots and the predicted curve behind them. The curve is flat at zero below an average degree of one and rises steeply above it.

The moment a giant appears

5 figures One piece, and connected, are different thresholds. Two curves against the average degree for graphs of 400 points: the largest component's share, rising from an average degree of one, and the probability of connectivity, rising only near the logarithm of the point count.

Two thresholds, not one

5 figures A triangle appears when the count says it should. The measured probability of containing a triangle against the edge chance, on graphs of 200 points, beside the expected number of triangles capped at one.

Finding a threshold with two moments

5 figures At the threshold the largest part is neither of the two obvious sizes. A table with one row per graph size, giving the largest component at the critical edge probability and that value divided by three candidate scalings.

The window where the giant is born

6 figures One threshold narrowing, one staying wide. Probability curves for connectivity and for containing a triangle, plotted against the edge probability as a multiple of each property's own threshold, at several graph sizes.

Sharp, or merely a threshold

6 figures 58.6% at 46 candidates, against 37% without the values. The chance of ending with the best candidate when the values are shown, against the number of candidates, for 10 sizes. It falls towards 0.5802 rather than towards 1/e.

When the numbers are shown

5 figures About the fourth-best, whatever the size of the field. The smallest expected rank achievable by an online rule, against the number of candidates, for 10 sizes. It rises to 3.8516 at 2500 candidates and its limit is 3.8695.

Giving up on the best

5 figures An online rule taking nine tenths of what an oracle takes. The share of the oracle's expected maximum secured by the best single threshold, and by the threshold at the median of the maximum, for 8 field sizes of independent uniform values.

Half of what an oracle takes

5 figures Up to 4 choices among 60, and the thresholds that nest. For 60 candidates in random order and 1, 2, 3, 4 acceptances, the chance of ending with the best among those accepted — 37.3%, 60.0%, 74.3%, 83.5% — and where along the sequence the rule starts accepting for each number of choices in hand.

The thresholds that nest

5 figures Add the odds from the end: start at event 5 of 12. The odds of 12 independent events with chances 1/k — the secretary problem, the sum of the odds from the end reaching one at event 5, and the chance of stopping on the last success from each starting point, highest at 39.6%.

Add the odds from the end

5 figures A 6-sided die rolled 6 to 40 times: some face missing, and the sum stopped early. Curves of the chance that some face of a die has not yet appeared against the number of rolls, with the inclusion–exclusion sum stopped after one, two and three terms drawn around the exact curve.

A sum stopped early still says something

5 figures