Tag: statistics
All the articles with the tag "statistics".
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Gaps between random points
Posted on:Drop n points uniformly on a circle. The gaps are exchangeable, the expected largest gap is exactly Hₙ/n, about ln n times the average, and its exact distribution is an alternating sum due to Fisher. The same formula is the ln n load imbalance of consistent hashing.
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The Ehrenfest urn
Posted on:N numbered balls in two urns, one moved at random per step, is the random walk on the corners of the N-cube. Its equilibrium is the binomial, the mean imbalance decays like (1 − 2/N)^t, and the expected time to return to the all-in-one-urn state is exactly 2^N.
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The arcsine law
Posted on:In a fair coin-tossing game the fraction of time one player is ahead is not concentrated near one half. Its exact distribution is u₂ₖ u₂ₙ₋₂ₖ, the limit is the U-shaped arcsine density, and with probability 1/2 one player leads for more than 85% of the game.
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Kelly criterion
Posted on:How much of the bankroll to stake on a repeated favorable bet: maximizing the expected logarithm gives f* = p − q/b, with closed forms for the growth rate and the drawdowns.
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German tank problem
Posted on:Estimating a population size N from the largest of k random serial numbers: m(1 + 1/k) − 1 is unbiased with error of order N/k. Allied statisticians used it on German tank production.
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Percolation
Posted on:Keep each edge of a grid with probability p and ask whether a path crosses the grid. The answer flips at a sharp threshold, exactly 1/2 on the square lattice by self-duality.
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Shattering and the VC dimension
Posted on:The VC dimension of a class of yes/no rules is the largest set of points it can label in every way. Past it, the labelings stop doubling, and that single integer governs learnability.
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The Hanson–Wright inequality
Posted on:A quadratic form in independent random variables concentrates around the trace: a Gaussian tail near the mean via the Frobenius norm, an exponential tail further out via the operator norm.