distributionsutveckling

Distributions, in probability theory and statistics, describe how the values of a random variable are spread over possible outcomes. A distribution can be discrete, continuous, or mixed, and it is often summarized by a probability function that assigns probabilities to outcomes or ranges of outcomes. The cumulative distribution function (CDF), F(x) = P(X ≤ x), encodes the distribution for all real numbers and is nondecreasing, right-continuous, and limits to 0 as x → −∞ and to 1 as x → ∞.

Discrete distributions specify probabilities for individual outcomes using a probability mass function (pmf). Continuous distributions use a probability density function (pdf) such that probabilities are obtained by integrating the density over intervals. For both types, the CDF is obtained by integrating or summing the density or mass function. Common discrete distributions include the Bernoulli, Binomial, and Poisson; common continuous distributions include the Uniform, Normal, Exponential, and Gamma.

Distributions are characterized by moments, such as the mean (expected value) and variance, which describe central tendency and dispersion. Other tools include the moment generating function and the characteristic function, which facilitate analysis of sums of independent variables and convergence properties. Key results such as the law of large numbers and the central limit theorem describe how distributions behave under aggregation and approximation.

Applications span statistical inference, hypothesis testing, Bayesian analysis, and simulations. In modeling uncertainty, choosing an appropriate distribution reflects beliefs about a process and enables quantification of risk and decision making. In functional analysis, the term distribution also denotes generalized functions that extend the notion of a function to broader linear functionals, a related but distinct concept.