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Bertrand's Paradox (probability)
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Louis François Bertrand, Joseph Bertrand introduced it in his work ''Calcul des probabilités'' (1889) as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the Possibility space, domain of possibilities is infinite. Bertrand's formulation of the problem The Bertrand paradox is generally presented as follows: Consider an equilateral triangle that is Inscribed figure, inscribed in a circle. Suppose a chord (geometry), chord of the circle is chosen at random. What is the probability that the Chord (geometry), chord is longer than a side of the triangle? Bertrand gave three arguments (each using the principle of indifference), all apparently valid yet yielding different results: # The "random endpoints" method: Choose two random points on the circumference of the circle and draw th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Interpretation
The classical definition of probability or classical interpretation of probability is identified with the works of Jacob Bernoulli and Pierre-Simon Laplace: This definition is essentially a consequence of the principle of indifference. If elementary events are assigned equal probabilities, then the probability of a disjunction of elementary events is just the number of events in the disjunction divided by the total number of elementary events. The classical definition of probability was called into question by several writers of the nineteenth century, including John Venn and George Boole. The frequency probability, frequentist definition of probability became widely accepted as a result of their criticism, and especially through the works of R.A. Fisher. The classical definition enjoyed a revival of sorts due to the general interest in Bayesian probability, because Bayesian methods require a prior probability distribution and the principle of indifference offers one source of suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand2-scatterplot
Bertrand may refer to: Places * Bertrand, Missouri, US * Bertrand, Nebraska, US * Bertrand, New Brunswick, Canada * Bertrand Township, Michigan, US * Bertrand, Michigan * Bertrand, Virginia, US * Bertrand Creek, state of Washington * Saint-Bertrand-de-Comminges Saint-Bertrand-de-Comminges (, literally ''Saint-Bertrand of Comminges''; Gascon language, Gascon: ''Sent Bertran de Comenge'') is a Communes of France, commune (municipality) and former episcopal see in the Haute-Garonne Departments of France, ..., France * Bertrand (1981–94 electoral district), in Quebec * Bertrand (electoral district), a provincial electoral district in Quebec Other * Bertrand (name) * ''Bertrand'' (steamboat), an 1865 steamboat that sank in the Missouri River * Bertrand Baudelaire, a fictional character in ''A Series of Unfortunate Events'' * Bertrand competition, an economic model where firms compete on price * Bertrand's theorem, a theorem in classical mechanics * Bertrand's po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematical Gazette
''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journal was established in 1894 by Edward Mann Langley as the successor to the ''Reports of the Association for the Improvement of Geometrical Teaching''. William John Greenstreet was its editor-in-chief for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha. Editors-in-chief The following persons are or have been editor-in-chief: Abstracting and indexing The journal is abstracted and indexed in EBSCO databases, Emerging Sources Citation Index, Scopus Scopus is a scientific abstract and citation database, launched by the academic publisher Elsevier as a competitor to older Web of Science in 2004. The ensuing competition between the two databases has been characterized as "intense" and is c ..., and zbMA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Chicago Press
The University of Chicago Press is the university press of the University of Chicago, a Private university, private research university in Chicago, Illinois. It is the largest and one of the oldest university presses in the United States. It publishes a wide range of academic titles, including ''The Chicago Manual of Style'', numerous academic journals, and advanced monographs in the academic fields. The press is located just south of the Midway Plaisance on the University of Chicago campus. One of its quasi-independent projects is the BiblioVault, a digital repository for scholarly books. History The University of Chicago Press was founded in 1890, making it one of the oldest continuously operating university presses in the United States. Its first published book was Robert F. Harper's ''Assyrian and Babylonian Letters Belonging to the Kouyunjik Collections of the British Museum''. The book sold five copies during its first two years, but by 1900, the University of Chicago Pr ... [...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] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scaling (geometry)
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions ( isotropically). The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc. More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edwin Jaynes
Edwin Thompson Jaynes (July 5, 1922 – April 30, 1998) was the Wayman Crow Distinguished Professor of Physics at Washington University in St. Louis. He wrote extensively on statistical mechanics and on foundations of probability and statistical inference, initiating in 1957 the maximum entropy interpretation of thermodynamics as being a particular application of more general Bayesian/information theory techniques (although he argued this was already implicit in the works of Josiah Willard Gibbs). Jaynes strongly promoted the interpretation of probability theory as an extension of logic. In 1963, together with his doctoral student Fred Cummings, he modeled the evolution of a two-level atom in an electromagnetic field, in a fully quantized way. This model is known as the Jaynes–Cummings model. A particular focus of his work was the construction of logical principles for assigning prior probability distributions; see the principle of maximum entropy, the principle of max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Physics
''Foundations of Physics'' is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedures". The journal publishes results and observations based on fundamental questions from all fields of physics, including: quantum mechanics, quantum field theory, special relativity, general relativity, string theory, M-theory, cosmology, thermodynamics, statistical physics, and quantum gravity ''Foundations of Physics'' has been published since 1970. Its founding editors were Henry Margenau and Wolfgang Yourgrau. The 1999 Nobel laureate Gerard 't Hooft was editor-in-chief from January 2007. At that stage, it absorbed the associated journal for shorter submissions ''Foundations of Physics Letters'', which had been edited by Alwyn Van der Merwe since its foundation in 1988. Past editorial board members (which include several Nobel lau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
