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Kaniadakis Weibull Distribution
The Kaniadakis Weibull distribution (or ''κ''-Weibull distribution) is a probability distribution arising as a generalization of the Weibull distribution. It is one example of a Kaniadakis ''κ''-distribution. The κ-Weibull distribution has been adopted successfully for describing a wide variety of complex systems in seismology, economy, epidemiology, among many others. Definitions Probability density function The Kaniadakis ''κ''-Weibull distribution is exhibits power-law right tails, and it has the following probability density function: : f_(x) = \frac \exp_\kappa(-\beta x^\alpha) valid for x \geq 0, where , \kappa, 0 is the scale parameter, and \alpha > 0 is the shape parameter or Weibull modulus. The Weibull distribution is recovered as \kappa \rightarrow 0. Cumulative distribution function The cumulative distribution function of ''κ''-Weibull distribution is given byF_\kappa(x) = 1 - \exp_\kappa(-\beta x^\alpha) valid for x \geq 0. The cumulative Weibul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rate Parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family of probability distributions is such that there is a parameter ''s'' (and other parameters ''θ'') for which the cumulative distribution function satisfies :F(x;s,\theta) = F(x/s;1,\theta), \! then ''s'' is called a scale parameter, since its value determines the " scale" or statistical dispersion of the probability distribution. If ''s'' is large, then the distribution will be more spread out; if ''s'' is small then it will be more concentrated. If the probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies :f_s(x) = f(x/s)/s, \! where ''f'' is the density of a standardized version of the density, i.e. f(x) \equiv f_(x). An estimator of a scale pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh Distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh (). A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Ra ... [...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] |
Probability Distributions
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes 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). 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 methodology, survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaniadakis Erlang Distribution
The Kaniadakis Erlang distribution (or κ-Erlang Gamma distribution) is a family of continuous statistical distributions, which is a particular case of the κ-Gamma distribution, when \alpha = 1 and \nu = n = positive integer. The first member of this family is the κ-exponential distribution of Type I. The κ-Erlang is a κ-deformed version of the Erlang distribution. It is one example of a Kaniadakis distribution. Characterization Probability density function The Kaniadakis ''κ''-Erlang distribution has the following probability density function: : f_(x) = \frac \prod_^n \left 1 + (2m -n)\kappa \rightx^ \exp_\kappa(-x) valid for x \geq 0 and n = \textrm \,\,\textrm , where 0 \leq , \kappa, < 1 is the entropic index associated with the Kaniadakis entropy. The ordinary [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaniadakis Gamma Distribution
The Kaniadakis Generalized Gamma distribution (or κ-Generalized Gamma distribution) is a four-parameter family of continuous statistical distributions, supported on a semi-infinite interval ,∞), which arising from the Kaniadakis statistics. It is one example of a Kaniadakis distribution. The κ-Gamma is a deformation of the Generalized gamma distribution">Generalized Gamma distribution. Definitions Probability density function The Kaniadakis ''κ''-Gamma distribution has the following probability density function: : f_(x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac \frac x^ \exp_\kappa(-\beta x^\alpha) valid for x \geq 0, where 0 \leq , \kappa, < 1 is the entropic index associated with the Kaniadakis entropy, , is the scale parameter, and is the shape parameter. The ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaniadakis Gaussian Distribution
The Kaniadakis Gaussian distribution (also known as ''κ''-Gaussian distribution) is a probability distribution which arises as a generalization of the Gaussian distribution from the maximization of the Kaniadakis entropy under appropriated constraints. It is one example of a Kaniadakis ''κ''-distribution. The κ-Gaussian distribution has been applied successfully for describing several complex systems in economy, geophysics, astrophysics, among many others. The κ-Gaussian distribution is a particular case of the κ-Generalized Gamma distribution. Definitions Probability density function The general form of the centered Kaniadakis ''κ''-Gaussian probability density function is: : f_(x) = Z_\kappa \exp_\kappa(-\beta x^2) where , \kappa, 0 is the scale parameter, and : Z_\kappa = \sqrt \Bigg( 1 + \frac\kappa \Bigg) \frac is the normalization constant. The standard Normal distribution is recovered in the limit \kappa \rightarrow 0. Cumulative distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giorgio Kaniadakis
Kaniadakis Giorgio ( el, Κανιαδάκης Γεώργιος; born on 5 June 1957 in Chania-Crete, Greece) a Greek-Italian physicist, is a Full Professor of Theoretical Physics at Politecnico di Torino (Italy) and is credited with introducing the concept oKaniadakis entropyand what is known as Kaniadakis statistics. He is in thWorld's Top 1% Scientists( Stanford University - Scopus Database), 2022. "Giorgio Kaniadakis has pioneered the surpassing of Boltzmann's Stosszahlansatz within special relativity and proposes a new entropy, emerging as the relativistic generalization of the Boltzmann entropy. Kaniadakis entropy generates power law-tailed distributions, which in the classical limit reduce to the Maxwell-Boltzmann exponential distribution" (From the Editorial by T. Biro of the volume Eur. Phys. J. A 40, N. 3, pp 255-256, 2009) Education In 1975 Giorgio Kaniadakis moved to Italy where he obtained the Bachelor's and Master's degrees in Nuclear Engineering in 1981 from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and evidence-based practice by identifying risk factors for disease and targets for preventive healthcare. Epidemiologists help with study design, collection, and statistical analysis of data, amend interpretation and dissemination of results (including peer review and occasional systematic review). Epidemiology has helped develop methodology used in clinical research, public health studies, and, to a lesser extent, basic research in the biological sciences. Major areas of epidemiological study include disease causation, transmission, outbreak investigation, disease surveillance, environmental epidemiology, forensic epidemiology, occupational epidemiology, screening, biomonitoring, and comparisons of treatment effects such as in clinical t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gutenberg–Richter Law
In seismology, the Gutenberg–Richter law (GR law) expresses the relationship between the magnitude and total number of earthquakes in any given region and time period of ''at least'' that magnitude. : \!\,\log_ N = a - b M or : \!\,N = 10^ where * \!\, N is the number of events having a magnitude \!\, \ge M , * \!\, a and \!\, b are constants, i.e. they are the same for all values of ''N'' and ''M''. Since magnitude is logarithmic, this is an instance of the Pareto distribution. The Gutenberg–Richter law is also widely used for acoustic emission analysis due to a close resemblance of acoustic emission phenomenon to seismogenesis. Background The relationship between earthquake magnitude and frequency was first proposed by Charles Francis Richter and Beno Gutenberg in a 1944 paper studying earthquakes in California, and generalised in a worldwide study in 1949. This relationship between event magnitude and frequency of occurrence is remarkably common, although the val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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