Probability Measures & Random Variables

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Abstractions about probability spaces, events, random variables, distributions, kernels, filtrations, and concentration bounds. They describe how uncertainty is represented, transformed, conditioned, compared, and approximated across empirical processes and Gaussian or generalized probabilistic systems.

36 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.
• Asymptotic equipartition property — The information-theoretic property that long source sequences concentrate on a typical set whose members have nearly equal exponential probability.
• Category of Markov kernels — A category whose objects are measurable spaces and whose morphisms are Markov kernels, composed by integrating one conditional probability kernel through another.
• Characteristic function (probability theory) — The Fourier–Stieltjes transform of a probability law, whose values uniquely determine the distribution.
• Complementary event — For an event in a sample space, the event containing exactly the outcomes in the sample space that are not in the original event.
• Complex random vector — A random element of a finite-dimensional complex vector space, equivalently a jointly distributed collection of complex-valued random variables.
• Dawson–Gärtner theorem — A projective-limit theorem lifting compatible finite-dimensional large-deviation principles to the inverse-limit space.
• Donsker classes — Classes of measurable functions for which the centered empirical process converges weakly in a uniform-function space to a tight Gaussian limit.
• Doob martingale — Track the evolving conditional expectation E[Y|F_t] of an integrable target as a filtration reveals information, producing a martingale of progressively refined best predictions.
• Dvoretzky–Kiefer–Wolfowitz inequality — A distribution-free exponential bound on the probability that an empirical cumulative distribution function deviates uniformly from its population distribution.
• Empirical process — A stochastic process indexing the centered and scaled difference between an empirical measure and its population expectation over a class of functions or sets.
• Energy distance — A metric between probability distributions built from expected pairwise Euclidean distances within and across independent samples.
• Event (probability theory) — A measurable subset of a sample space representing the collection of outcomes for which a probabilistic proposition holds.
• 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.
• Gaussian process emulator — A probabilistic surrogate that uses a Gaussian process fitted to selected simulator runs to predict an expensive model's output and quantify interpolation uncertainty.
• Giry monad — The probability monad on measurable spaces that sends each space to its measurable space of probability measures.
• 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.
• Law of total covariance — The identity decomposing covariance into expected conditional covariance plus covariance of conditional expectations.
• Law of total probability — A probability identity expressing an event's probability as the sum or integral of its conditional probabilities over a mutually exclusive exhaustive partition.
• Location–scale family — A family of probability distributions closed under positive affine transformations of a fixed standardized random variable.
• Markov operator — A positive mass-preserving operator that propagates probability densities, measures or observables through a stochastic transition.
• 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.
• Modified half-normal distribution — A positive-support probability family extending the half-normal shape with power and exponential-tilt parameters.
• Natural filtration — The smallest filtration that makes a given stochastic process adapted by recording exactly the events observable from its history up to each time.
• Outcome (probability) — One elementary possible result of a random experiment, represented as a single element of its sample space.
• Postselection — Conditioning an experiment, probability model or computation on a specified event after outcomes are available, thereby replacing the original distribution with its conditional distribution.
• Probability axioms — The foundational conditions requiring a probability measure to be nonnegative, assign one to the whole sample space and add over countably many disjoint events.
• Probability measure — A countably additive measure on a sigma-algebra that assigns total mass one to the sample space.
• Ratio distribution — The probability distribution of a random variable formed as the quotient of two random variables.
• Reciprocal distribution — A bounded positive distribution whose density is proportional to one over the variable, equivalently uniform after logarithmic transformation.
• Truncated normal distribution — A normal probability law conditioned to lie within a specified lower, upper, or two-sided interval.
• Unit measure — The probability axiom requiring the measure of the entire sample space to equal one.
• Van den Berg–Kesten inequality — A product-measure inequality bounding the probability of disjoint occurrence of two events by the product of their individual probabilities.