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Stability (probability)
In probability theory, the stability of a random variable is the property that a linear combination of two independent copies of the variable has the same distribution, up to location and scale parameters. The distributions of random variables having this property are said to be "stable distributions". Results available in probability theory show that all possible distributions having this property are members of a four-parameter family of distributions. The article on the stable distribution describes this family together with some of the properties of these distributions. The importance in probability theory of "stability" and of the stable family of probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed random variables. Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution. For details see stable distribution. Definition There are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univariate Distribution
In statistics, a univariate distribution is a probability distribution of only one random variable. This is in contrast to a multivariate distribution, the probability distribution of a random vector (consisting of multiple random variables). Examples One of the simplest examples of a discrete univariate distribution is the discrete uniform distribution, where all elements of a finite set are equally likely. It is the probability model for the outcomes of tossing a fair coin, rolling a fair die, etc. The univariate continuous uniform distribution on an interval 'a'', ''b''has the property that all sub-intervals of the same length are equally likely. Other examples of discrete univariate distributions include the binomial, geometric, negative binomial, and Poisson distributions.Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Discrete Univariate Distributions, 3rd Edition, Wiley, . At least 750 univariate discrete distributions have been reported in the literature.Wimmer G, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indecomposable Distribution
In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: ''Z'' ≠ ''X'' + ''Y''. If it can be so expressed, it is decomposable: ''Z'' = ''X'' + ''Y''. If, further, it can be expressed as the distribution of the sum of two or more independent ''identically'' distributed random variables, then it is divisible: ''Z'' = ''X''1 + ''X''2. Examples Indecomposable * The simplest examples are Bernoulli-distributions: if ::X = \begin 1 & \text p, \\ 0 & \text 1-p, \end :then the probability distribution of ''X'' is indecomposable. :Proof: Given non-constant distributions ''U'' and ''V,'' so that ''U'' assumes at least two values ''a'', ''b'' and ''V'' assumes two values ''c'', ''d,'' with ''a'' < ''b'' and ''c'' < ''d'', then ''U'' + ''V'' as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
