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Subexponential Distribution (light-tailed)
In probability theory, one definition of a subexponential distribution is as a probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ... whose tails decay at an exponential rate, or faster: a real-valued distribution \cal D is called subexponential if, for a random variable X\sim , :(, X, \ge x)=O(e^) , for large x and some constant K>0. The subexponential norm, \, \cdot\, _, of a random variable is defined by :\, X\, _:=\inf\ \, where the infimum is taken to be +\infty if no such K exists. This is an example of a Orlicz norm. An equivalent condition for a distribution \cal D to be subexponential is then that \, X\, _0. # (, X, ^p)^\le K p, for all p\ge 1 and some constant K>0. # For some constant K>0, (e^) \le e^ for all 0\le \lambda \le 1/K. # (X) exists ... [...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] |
Probability Distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical description of a Randomness, random phenomenon in terms of its sample space and the Probability, probabilities of Event (probability theory), events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that fair coin, the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orlicz Space
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the ''L''''p'' spaces. Like the ''L''''p'' spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932. Besides the ''L''''p'' spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space ''L'' log+ ''L'', which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions ''f'' such that the :\int_ , f(x), \log^+ , f(x), \,dx , is a Young function, i.e. Convex function">convex, lower semicontinuous, and non-trivial, in the sense that it is not the zero function x \mapsto 0 , and it is not the convex dual of the zero function x \mapsto \begin 0 \text x = 0, \\ +\infty \text\end Orlicz spaces Let L^\dagger_\Phi be the set of measurable functions ''f'' : ''X'' → R such that the inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sub-Gaussian
In probability theory, a subgaussian distribution, the distribution of a subgaussian random variable, is a probability distribution with strong tail decay. More specifically, the tails of a subgaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives subgaussian distributions their name. Often in analysis, we divide an object (such as a random variable) into two parts, a central bulk and a distant tail, then analyze each separately. In probability, this division usually goes like "Everything interesting happens near the center. The tail event is so rare, we may safely ignore that." Subgaussian distributions are worthy of study, because the gaussian distribution is well-understood, and so we can give sharp bounds on the rarity of the tail event. Similarly, the subexponential distributions are also worthy of study. Formally, the probability distribution of a random variable ''X '' is called subgaussian if there is a posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
