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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] |
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] |
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] |
Infinitely Divisible Probability Distributions
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter, space, time, money, or abstract mathematical objects such as the continuum. In philosophy The origin of the idea in the Western tradition can be traced to the 5th century BCE starting with the Ancient Greek pre-Socratic philosopher Democritus and his teacher Leucippus, who theorized matter's divisibility beyond what can be perceived by the senses until ultimately ending at an indivisible atom. The Indian philosopher, Maharshi Kanada also proposed an atomistic theory, however there is ambiguity around when this philosopher lived, ranging from sometime between the 6th century to 2nd century BCE. Around 500 BC, he postulated that if we go on dividing matter ('' padarth''), we shall get smaller and smaller particles. Ultimately, a time ... [...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 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] |
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] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less ... [...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] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaniadakis Statistics
Kaniadakis statistics (also known as κ-statistics) is a generalization of Boltzmann-Gibbs statistical mechanics, based on a relativistic generalization of the classical Boltzmann-Gibbs-Shannon entropy (commonly referred to as Kaniadakis entropy or κ-entropy). Introduced by the Greek Italian physicist Giorgio Kaniadakis in 2001, κ-statistical mechanics preserve the main features of ordinary statistical mechanics and have attracted the interest of many researchers in recent years. The κ-distribution is currently considered one of the most viable candidates for explaining complex physical, natural or artificial systems involving power-law tailed statistical distributions. Kaniadakis statistics have been adopted successfully in the description of a variety of systems in the fields of cosmology, astrophysics, condensed matter, quantum physics, seismology, genomics, economics, epidemiology, and many others. Mathematical formalism The mathematical formalism of κ-statistics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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