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Singular Distribution
A singular distribution or singular continuous distribution is a probability distribution concentrated on a set of Lebesgue measure zero, for which the probability of each point in that set is zero. Properties Such distributions are not absolutely continuous with respect to Lebesgue measure. A singular distribution is not a discrete probability distribution because each discrete point has a zero probability. On the other hand, neither does it have a probability density function, since the Lebesgue integral of any such function would be zero. In general, distributions can be described as a discrete distribution (with a probability mass function), an absolutely continuous distribution (with a probability density), a singular distribution (with neither), or can be decomposed into a mixture of these. Example An example is the Cantor distribution; its cumulative distribution function is a devil's staircase. Less curious examples appear in higher dimensions. For example, the ... [...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] |
Null Set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = (X, \Sigma, \mu) a null set is a set S \in \Sigma such that \mu(S) = 0. Examples Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers , the set of rational numbers and the set of algebraic numbers are all countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Continuous
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : ''absolutely continuous'' ⊆ '' u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Probability Distribution
In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of 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 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 probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Distribution
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution. Its cumulative distribution function is continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the Devil's staircase, although that term has a more general meaning. Characterization The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets: : \begin C_0 = & ,1\\ pt C_1 = & ,1/3cup /3,1\\ pt C_2 = & ,1/9c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Function
In mathematics, the Cantor function is an example of a function (mathematics), function that is continuous function, continuous, but not absolute continuity, absolutely continuous. It is a notorious Pathological_(mathematics)#Pathological_example, counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and Measure (mathematics), measure. Although it is continuous everywhere, and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument goes from 0 to 1. Thus, while the function seems like a constant one that cannot grow, it does indeed Monotonic function, monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Copula (probability Theory)
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval , 1 Copulas are used to describe / model the dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but only metaphoricly related to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Measure
In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on all measurable subsets of B while \nu is zero on all measurable subsets of A. This is denoted by \mu \perp \nu. A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples. Examples on R''n'' As a particular case, a measure defined on the Euclidean space \R^n is called ''singular'', if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure. Example. A discrete measure. The Heaviside step function on the real line, H(x) \ \stackrel \begin 0, & x 0 but \delta_0(U) = 0. Example. A singular continuous measure. The Cantor distribution has a cumulative dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue's Decomposition Theorem
In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem provides a way to decompose a measure into two distinct parts based on their relationship with another measure. Definition The theorem states that if (\Omega,\Sigma) is a measurable space and \mu and \nu are σ-finite signed measures on \Sigma, then there exist two uniquely determined σ-finite signed measures \nu_0 and \nu_1 such that: * \nu=\nu_0+\nu_1\, * \nu_0\ll\mu (that is, \nu_0 is absolutely continuous with respect to \mu) * \nu_1\perp\mu (that is, \nu_1 and \mu are singular). Refinement Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue-Radon-Nikodym theorem. That is, let (\Omega,\Sigma) be a measure space, \mu a σ-finite positive measure on \Sigma and \lambda a complex measure on \Sigma. * There is a unique pair of complex measures on \Sigma such that \lambda = \lambda_a + \lambda_s, \quad \lambda_a \ll \mu, \quad \lambda_s \perp \mu. If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
