Probability Theory¶

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44 domain-specific abstractions whose origin domain is Probability Theory.

Algebra of random variables — The symbolic calculus for forming functions of random variables and deriving the resulting distributions, moments and dependence-sensitive identities.
Branching process — A stochastic population process in which individuals independently generate random descendants according to a declared reproduction law.
Characteristic function (probability theory) — The Fourier–Stieltjes transform of a probability law, whose values uniquely determine the distribution.
Collectively exhaustive events — Require a declared family of events to cover the entire sample space, so every possible outcome belongs to at least one member without requiring the members to be disjoint.
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.
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.
Diffusion Process — A continuous-path continuous-time Markov process governed locally by drift and covariance and globally by its transition law.
Discrete-time Markov chain — A stochastic sequence whose next-state distribution depends on the current state and transition step but not on the earlier path once the present is known.
Dispersion Function — The convex location-indexed functional \(D_X(u)=\mathbb{E}|X-u|\), whose slopes recover an integrable real distribution.
Doob–Dynkin lemma — Characterize when a measurable quantity carries no information beyond a measurable map: under the stated measurable-space conditions it factors as a measurable function of that map exactly when it is measurable with respect to the map's generated sigma-algebra.
Dvoretzky–Kiefer–Wolfowitz inequality — A distribution-free exponential bound on the probability that an empirical cumulative distribution function deviates uniformly from its population distribution.
Event (probability theory) — A measurable subset of a sample space representing the collection of outcomes for which a probabilistic proposition holds.
Factorial moment — Take the expectation of a random variable’s falling factorial to expose counting structure and probability-generating-function derivatives.
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.
Inverse distribution — The probability distribution of the reciprocal of a random 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.
Location–scale family — A family of probability distributions closed under positive affine transformations of a fixed standardized random variable.
Lévy–Prokhorov metric — A distance between probability measures on a metric space that permits both spatial enlargement and a matching probability slack, metrizing weak convergence on separable spaces and connecting compactness to tightness.
Markov kernel — A measurable assignment sending each source point to a probability measure on a target space, generalizing a stochastic transition matrix to arbitrary measurable spaces.
Markov operator — A positive mass-preserving operator that propagates probability densities, measures or observables through a stochastic transition.
Martingale (probability theory) — Model an adapted integrable stochastic process whose conditional expected future value, given present information, equals its current value.
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.
Mixture Distribution — A probability law generated by first selecting a latent component according to normalized weights and then sampling from that component.
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.
Negative Hypergeometric Distribution — The distribution of failures observed before a fixed number of successes when sampling without replacement from a finite population.
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 integral transform — The result that applying a continuous random variable's own cumulative distribution function produces a standard uniform random variable.
Probability measure — A countably additive measure on a sigma-algebra that assigns total mass one to the sample space.
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.
Reflection principle (Wiener process) — Reflect a Wiener path after its first hitting time of a level to obtain another process with the same law, converting barrier-crossing events into endpoint-distribution identities.
Renewal theory — A probability framework for processes that restart after independent identically distributed waiting times, studying event counts, ages, residual lives and rewards over repeated cycles.
Sub-probability measure — A nonnegative countably additive measure whose total mass is at most one, allowing missing mass to represent termination, failure or an unmodeled outcome.
Truncated normal distribution — A normal probability law conditioned to lie within a specified lower, upper, or two-sided interval.
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.
Variance-gamma distribution — A continuous heavy-tailed probability family obtained by evaluating Brownian motion with drift at an independent gamma-distributed random time, equivalently a normal variance-mean gamma mixture.
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.
Σ-Algebra of τ-past — The stopped sigma-algebra Fτ containing exactly those events whose truth is knowable by a stopping time τ, defined by compatibility of each event with {τ≤t} and the filtration Ft.