Distributions

Distributions is a term used in mathematics and statistics with two closely related but distinct meanings. In probability and statistics, a distribution describes how random outcomes are allocated, via a probability measure, probability mass function for discrete variables or probability density function for continuous variables. The cumulative distribution function F gives P(X ≤ x). Common discrete distributions include Binomial, Poisson, and Geometric; common continuous ones include Normal, Exponential, Uniform, Gamma, and Beta. Distributions are characterized by parameters, moments, tails, and support, and they underlie statistical inference, hypothesis testing, and modeling. Limit theorems such as the law of large numbers and central limit theorem describe how sample distributions converge to limiting ones.

In analysis, distributions (also called generalized functions) generalize functions to rigorously treat objects like the Dirac delta. A distribution is a continuous linear functional on a space of test functions, typically smooth functions with compact support. Key examples include the Dirac delta δ and the Heaviside step function. Distributions can be differentiated, multiplied by smooth functions, and convolved with test functions. They extend classical derivatives and Fourier transforms to nonsmooth or non-integrable objects and are foundational in partial differential equations, mathematical physics, and signal processing. The space of tempered distributions, dual to the Schwartz space, permits Fourier transforms of a broad class of objects.

Connections between probability and distribution theory arise through characteristic functions and Fourier transforms, which encode distributions as transforms. The concept also assists in treating limiting processes that do not yield ordinary densities. Distribution theory was developed by Laurent Schwartz in the 20th century and has since become standard in analysis and mathematical physics.