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Intensity Measure
In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random measure of a set, hence it corresponds to the average volume the random measure assigns to a set. The intensity measure contains important information about the properties of the random measure. A Poisson point process, interpreted as a random measure, is for example uniquely determined by its intensity measure. Definition Let \zeta be a random measure on the measurable space (S, \mathcal A) and denote the expected value of a random element Y with \operatorname E . The intensity measure : \operatorname E \zeta \colon \mathcal A \to ,\infty of \zeta is defined as : \operatorname E \zeta(A)= \operatorname E zeta(A) for all A \in \mathcal A. Note the difference in notation between the expectation value of a random element Y , denoted by \operatorname E and the intensity ... [...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] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Measure
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. Definition Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let E be a separable complete metric space and let \mathcal E be its Borel \sigma -algebra. (The most common example of a separable complete metric space is \R^n .) As a transition kernel A random measure \zeta is a ( a.s.) locally finite transition kernel from an abstract probability space (\Omega, \mathcal A, P) to (E, \mathcal E) . Being a transition kernel means that *For any fixed B \in \mathcal \mathcal E , the mapping : \omega \mapsto \zeta(\omega,B) :is measurable from (\Omega, \mathcal A) to (\R, \mathcal B(\R)) *For every fixed \omega \in \Omega , the mapping : B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as \mathbb or . Histo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Point Process
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, and volume with a set X of 'points' in the space, but ''regions'' of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region. Definition Consider a set X and a σ-algebra \mathcal F on X. Then the tuple (X, \mathcal F) is called a measurable space. The elements of \mathcal F are called measurable sets within the measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measures (measure Theory)
Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England measure, legislation of the Church of England * Measure of the National Assembly for Wales, primary legislation in Wales between 1999 and 2011 * Assembly Measure of the Northern Ireland Assembly (1973) Science and mathematics * Measure (data warehouse), a property on which calculations can be made * Measure (mathematics), a systematic way to assign a number to each suitable subset of a given set * Measure (physics), a way to integrate over all possible histories of a system in quantum field theory * Measure (termination), in computer program termination analysis * Measuring coalgebra, a coalgebra constructed from two algebras * Measure (Apple), an iOS augmented reality app Other uses * ''Measure'' (album), by Matt Pond PA, 2000, and its title track * Measure (bartending) or ji ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
