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Holtsmark Distribution
The (one-dimensional) Holtsmark distribution is a continuous probability distribution. The Holtsmark distribution is a special case of a stable distribution with the index of stability or shape parameter \alpha equal to 3/2 and the skewness parameter \beta of zero. Since \beta equals zero, the distribution is symmetric, and thus an example of a symmetric alpha-stable distribution. The Holtsmark distribution is one of the few examples of a stable distribution for which a closed form expression of the probability density function is known. However, its probability density function is not expressible in terms of elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions. The Holtsmark distribution has applications in plasma physics and astrophysics. In 1919, Norwegian physicist Johan Peter Holtsmark proposed the distribution as a model for the fluctuating fields in plasma due to the motion of charged particles. It is also appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levy DistributionPDF
Levy, Lévy or Levies may refer to: People * Levy (surname), people with the surname Levy or Lévy * Levy Adcock (born 1988), American football player * Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker * Levy Fidelix (1951–2021), Brazilian conservative politician, businessman and journalist * Levy Gerzberg (born 1945), Israeli-American entrepreneur, inventor, and business person * Levy Li (born 1987), Miss Malaysia Universe 2008–2009 * Levy Mashiane (born 1996), South African footballer * Levy Matebo Omari (born 1989), Kenyan long-distance runner * Levy Mayer (1858–1922), American lawyer * Levy Middlebrooks (born 1966), American basketball player * Levy Mokgothu, South African footballer * Levy Mwanawasa (1948–2008), President of Zambia from 2002 * Levy Nzoungou (born 1998), Congolese-French rugby player, playing in England * Levy Rozman (born 1995), American chess IM, coach, and content creator * Levy Sekgapane (born 1990), South Afri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. Finite data set of numbers The median of a finite list of numbers is the "middle" number, when those numbers are list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Laws
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Empirical examples The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, volcanic eruptions, human judgments of stimulus intensity and many other qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distributions With Non-finite Variance
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Distributions
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of Continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the coni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Nottingham
The University of Nottingham is a public university, public research university in Nottingham, United Kingdom. It was founded as University College Nottingham in 1881, and was granted a royal charter in 1948. The University of Nottingham belongs to the research intensive Russell Group association. Nottingham's main campus (University Park Campus, Nottingham, University Park) with Jubilee Campus and teaching hospital (Queen's Medical Centre) are located within the City of Nottingham, with a number of smaller campuses and sites elsewhere in Nottinghamshire and Derbyshire. Outside the UK, the university has campuses in Semenyih, Malaysia, and Ningbo, China. Nottingham is organised into five constituent faculties, within which there are more than 50 schools, departments, institutes and research centres. Nottingham has about 45,500 students and 7,000 staff, and had an income of £694 million in 2020–21, of which £114.9 million was from research grants and contracts. The institution's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile."van der Waals profile" appears with lowercase "van" in almost all sources, such as: ''Statistical mechanics of the liquid surface'' by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, , and in ''Journal of technical physics'', Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995/ref> It is a special case of the inverse-gamma distribution. It is a stable distribution. Definition The probability density function of the Lévy distribution over the domain x\ge \mu is :f(x;\mu,c)=\sqrt~~\frac where \mu is the location parameter and c is the scale parameter. The cumulative distribution function is :F(x;\mu,c)=1 - \te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f(x; x_0,\gamma) is the distribution of the -intercept of a ray issuing from (x_0,\gamma) with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero. The Cauchy distribution is often used in statistics as the canonical example of a " pathological" distribution since both its expected value and its variance are undefined (but see below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist., Chapter 16. The Cauchy distribution has no moment generating function. In mathematics, it is closely related to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes 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). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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