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Stable Distribution
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. Of the four parameters defining the family, most attention has been focused on the stability parameter, \alpha (see panel). Stable distributions have 0 < \alpha \leq 2, with the upper bound corresponding to the , and to the Cauchy distribution. The distributio ... [...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 African ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose. The ''arithmetic mean'', also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the '' sample mean'' (\bar) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted \mu or \mu_x. Outside probability and statistics, a wide rang ... [...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 Support (measure theory), supported on the real numbers, discrete or "mixed" as well as Continuous variable, continuous, is uniquely identified by a right-continuous Monotonic function, monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow [0,1] 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 negative 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Moment
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location. Sets of central moments can be defined for both univariate and multivariate distributions. Univariate moments The -th moment about the mean (or -th central moment) of a real-valued random variable is the quantity , where E is the expectation operator. For a continuous univariate probability distribution with probability density ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution ( Hausdorff moment problem). The same is not true on unbounded intervals ( Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the moments of random variables. Significance of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign Function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as \sgn x or \sgn (x). Definition The signum function of a real number x is a piecewise function which is defined as follows: \sgn x :=\begin -1 & \text x 0. \end The law of trichotomy states that every real number must be positive, negative or zero. The signum function denotes which unique category a number falls into by mapping it to one of the values , or which can then be used in mathematical expressions or further calculations. For example: \begin \sgn(2) &=& +1\,, \\ \sgn(\pi) &=& +1\,, \\ \sgn(-8) &=& -1\,, \\ \sgn(-\frac) &=& -1\,, \\ \sgn(0) &=& 0\,. \end Basic properties Any real number can be expressed as the product of its absolute value and its sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (probability Theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions, characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the charact ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degenerate Distribution
In probability theory, a degenerate distribution on a measure space (E, \mathcal, \mu) is a probability distribution whose support is a null set with respect to \mu. For instance, in the -dimensional space endowed with the Lebesgue measure, any distribution concentrated on a -dimensional subspace with is a degenerate distribution on . This is essentially the same notion as a singular probability measure, but the term ''degenerate'' is typically used when the distribution arises as a limit of (non-degenerate) distributions. When the support of a degenerate distribution consists of a single point , this distribution is a Dirac measure in : it is the distribution of a deterministic random variable equal to with probability 1. This is a special case of a discrete distribution; its probability mass function equals 1 in and 0 everywhere else. In the case of a real-valued random variable, the cumulative distribution function of the degenerate distribution localized in is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Journal Of Political Economy
The ''Journal of Political Economy'' is a monthly peer-reviewed academic journal published by the University of Chicago Press. Established by James Laurence Laughlin in 1892, it covers both theoretical and empirical economics. In the past, the journal published quarterly from its introduction through 1905, ten issues per volume from 1906 through 1921, and bimonthly from 1922 through 2019. The editor-in-chief is Esteban Rossi-Hansberg (University of Chicago). It is considered one of the top five journals in economics. JPE Micro and JPE Macro In 2023, University of Chicago Press announced the establishment of Journal of Political Economy Microeconomics (JPE Micro) and Journal of Political Economy Macroeconomics (JPE Macro), two new journals that are vertically integrated with the Journal of Political Economy. Abstracting and indexing The journal is abstracted and indexed in EBSCO, ProQuest, EconLit, Research Papers in Economics, Current Contents/Social & Behavioral Scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vilfredo Pareto
Vilfredo Federico Damaso Pareto (; ; born Wilfried Fritz Pareto; 15 July 1848 – 19 August 1923) was an Italian polymath, whose areas of interest included sociology, civil engineering, economics, political science, and philosophy. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices, and was one of the minds behind the Lausanne School of economics. He was also responsible for popularising the use of the term '' elite'' in social analysis and contributed to elite theory. He has been described as "one of the last Renaissance scholars. Trained in physics and mathematics, he became a polymath whose genius radiated into nearly all other major fields of knowledge." He introduced the concept of Pareto efficiency and helped develop the field of microeconomics. He was also the first to claim that income follows a Pareto distribution, which is a power law probability distribution. The Paret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
