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Hyperbolic Secant Distribution
In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution. Generalisation of the distribution gives rise to the Meixner distribution, also known as the Natural Exponential Family - Generalised Hyperbolic Secant or NEF-GHS distribution. Explanation A random variable follows a hyperbolic secant distribution if its probability density function (pdf) can be related to the following standard form of density function by a location and shift transformation: :f(x) = \frac12 \; \operatorname\!\left(\frac\,x\right)\! , where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyper Secant Pdf
Hyper may refer to: Arts and entertainment * ''Hyper'' (2016 film), 2016 Indian Telugu film * ''Hyper'' (2018 film), 2018 Indian Kannada film * ''Hyper'' (magazine), an Australian video game magazine *Hyper (TV channel), a Filipino sports channel * Hyper+, a former Polish programming block on Teletoon+ Mathematics * Hypercube, the n-dimensional analogue of a square and a cube *Hyperoperation, an arithmetic operation beyond exponentiation * Hyperplane, a subspace whose dimension is one less than that of its ambient space *Hypersphere, the set of points at a constant distance from a given point called its centre *Hypersurface, a generalization of the concepts of hyperplane, plane curve, and surface *Hyperstructure, an algebraic structure equipped with at least one multivalued operation *Hyperbolic functions, analogues of trigonometric functions defined using the hyperbola rather than the circle Other uses *DJ Hyper (born 1977), a British electronic musician *Hyper key, a modifier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. 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 or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate Analytics, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed subject sectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Exponential Family
In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF). Definition Univariate case The natural exponential families (NEF) are a subset of the exponential families. A NEF is an exponential family in which the natural parameter ''η'' and the natural statistic ''T''(''x'') are both the identity. A distribution in an exponential family with parameter ''θ'' can be written with probability density function (PDF) : f_X(x\mid \theta) = h(x)\ \exp\Big(\ \eta(\theta) T(x) - A(\theta)\ \Big) \,\! , where h(x) and A(\theta) are known functions. A distribution in a natural exponential family with parameter θ can thus be written with PDF : f_X(x\mid \theta) = h(x)\ \exp\Big(\ \theta x - A(\theta)\ \Big) \,\! . [Note that slightly different notation is used by the originator of the NEF, Carl Morris.Morris C. (2006) "Natural exponential families", ''Encyclopedia of Stati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josef Meixner
Josef Meixner (24 April 1908 – 19 March 1994) was a German theoretical physicist, known for his work on the physics of deformable bodies, thermodynamics, statistical mechanics, Meixner polynomials, Meixner–Pollaczek polynomials, and spheroidal wave functions.Meixner – CIO NIST Education Meixner began his studies in theoretical physics with Arnold Sommerfeld at the Ludwig Maximilian University of Munich in 1926. He was awarded his doctorate in 1931, with the submission of a thesis on the application of the Green's function, Green function in quantum mechanics.Career Meixner taught at a high school for a few years. He was an assistant at the Institute of Theoretical Physics in Munchen until 1934. He worked with Salomon Bochner to determine that the Hermite polynomials were ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Language
The Wolfram Language ( ) is a general multi-paradigm programming language developed by Wolfram Research. It emphasizes symbolic computation, functional programming, and rule-based programming and can employ arbitrary structures and data. It is the programming language of the mathematical symbolic computation program Mathematica. History The Wolfram Language was a part of the initial version of Mathematica in 1988. Symbolic aspects of the engine make it a computer algebra system. The language can perform integration, differentiation, matrix manipulations, and solve differential equations using a set of rules. Also, the initial version introduced the notebook model and the ability to embed sound and images, according to Theodore Gray's patent. Wolfram also added features for more complex tasks, such as 3D modeling. A name was finally adopted for the language in 2013, as Wolfram Research decided to make a version of the language engine free for Raspberry Pi users, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal distribution, negative skew commonly indicates that the ''tail'' is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution, but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but fat. Introduction Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are called ''tai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. If X_1, X_2, \dots, X_n, \dots are random samples drawn from a population with overall mean \mu and finite variance and if \bar_n is the sample mean of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Identically Distributed Random Variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independence (probability theory), independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Distribution
Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, the inverse of the logistic function ** Logistic distribution, the derivative of the logistic function, a continuous probability distribution, used in probability theory and statistics * Mathematical logic, subfield of mathematics exploring the applications of formal logic to mathematics Other uses * Logistics, the management of resources and their distributions ** Logistic engineering, the scientific study of logistics ** Military logistics Military logistics is the discipline of planning and carrying out the movement, supply, and maintenance of military forces. In its most comprehensive sense, it is those aspects or military operations that deal with: * Design, development, acqui ..., the study of logistics at the service of milit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
