Probability Distributions & Transforms

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Abstractions about characterizing and transforming probability distributions — specific distribution families (hyperbolic, zeta, truncated normal, Johnson's SU), transform and summary functions (characteristic function, cumulative distribution function, quantile function), and results relating random variables to their derived distributions (ratio distribution, probability integral transform).

22 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Algebra of random variables — The symbolic calculus for forming functions of random variables and deriving the resulting distributions, moments and dependence-sensitive identities.
• Characteristic function (probability theory) — The Fourier–Stieltjes transform of a probability law, whose values uniquely determine the distribution.
• Cumulative distribution function — For a real-valued random variable X, the function F(x)=P(X≤x), a nondecreasing right-continuous map whose limits are zero and one and which uniquely determines the distribution.
• Dawson–Gärtner theorem — A projective-limit theorem lifting compatible finite-dimensional large-deviation principles to the inverse-limit space.
• Dirac delta function — A distribution concentrated at one point whose action on a test function returns that function’s value at the point.
• Dvoretzky–Kiefer–Wolfowitz inequality — A distribution-free exponential bound on the probability that an empirical cumulative distribution function deviates uniformly from its population distribution.
• Extreme value theory — A branch of statistics modeling the limiting behavior and tail risk of unusually large or small observations, especially block maxima and threshold exceedances.
• Factorial moment — Take the expectation of a random variable’s falling factorial to expose counting structure and probability-generating-function derivatives.
• False confidence theorem — Show that a continuous data-dependent additive probability distribution can, for some false assertion, assign arbitrarily high belief with high sampling probability, motivating assertion-wise validity checks.
• Gaussian probability space — A probability space equipped with a closed Hilbert subspace of centered real Gaussian random variables, optionally separated from a transverse sigma-algebra.
• Hyperbolic distribution — A continuous probability distribution whose log-density traces a hyperbola, yielding exponential but heavier-than-normal tails.
• Inverse distribution — The probability distribution of the reciprocal of a random variable.
• Johnson's SU-distribution — An unbounded four-parameter distribution obtained by applying an inverse-hyperbolic-sine transformation to a standard normal variable.
• Large deviations of Gaussian random functions — The asymptotic study of rare high excursions of Gaussian processes or fields over large domains or thresholds.
• McDiarmid's inequality — A concentration bound for a function of independent variables whose value can change by at most cᵢ when only coordinate i is replaced.
• Probability box — A pair of noncrossing lower and upper cumulative-distribution bounds representing a set of admissible probability distributions for an uncertain quantity.
• Probability integral transform — The result that applying a continuous random variable's own cumulative distribution function produces a standard uniform random variable.
• Quantile function — A generalized inverse of a cumulative distribution function that maps a probability level to the smallest value whose cumulative probability reaches that level.
• Ratio distribution — The probability distribution of a random variable formed as the quotient of two random variables.
• Stochastic ordering — A partial-order comparison of probability distributions stating that one is larger than another according to a declared class of increasing tests or risk criteria.
• Truncated normal distribution — A normal probability law conditioned to lie within a specified lower, upper, or two-sided interval.
• Zeta distribution — A discrete power-law distribution on positive integers with probability proportional to k^−s and normalized by the Riemann zeta function for s>1.