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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] |
Levy0 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] |
Moment Generating Function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of 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 moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions. As its name implies, the moment-generating function can be used to compute a distribution’s moments: the ''n''th moment about 0 is the ''n''th derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. The moment-generating func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Transform Sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, Smirnov transform, or the golden ruleAalto University, N. Hyvönen, Computational methods in inverse problems. Twelfth lecture https://noppa.tkk.fi/noppa/kurssi/mat-1.3626/luennot/Mat-1_3626_lecture12.pdf) is a basic method for pseudo-random number sampling, i.e., for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u between 0 and 1, interpreted as a probability, and then returns the largest number x from the domain of the distribution P(X) such that P(-\infty , e.g. from U \sim \mathrm ,1 #Find the inverse of the desired CDF, e.g. F_X^(x). # Compute X=F_X^(u). The computed random variable X has distribution F_X(x). Expressed differently, given a continuous uniform variable U in ,1/math> and an invertible cumu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folded Normal Distribution
The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called "folded" because probability mass to the left of ''x'' = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. Definitions Density The probability density function (PDF) is given by :f_Y(x;\mu,\sigma^2)= \frac \, e^ + \frac \, e^ for ''x'' ≥ 0, and 0 everywhere else. An alternative formulation is given by : f\left(x \right)=\sqrte^\cosh, where cosh is the cosine Hyperbolic function. It foll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scaled-inverse-chi-squared Distribution
The scaled inverse chi-squared distribution is the distribution for ''x'' = 1/''s''2, where ''s''2 is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ2 = τ2. The distribution is therefore parametrised by the two quantities ν and τ2, referred to as the ''number of chi-squared degrees of freedom'' and the ''scaling parameter'', respectively. This family of scaled inverse chi-squared distributions is closely related to two other distribution families, those of the inverse-chi-squared distribution and the inverse-gamma distribution. Compared to the inverse-chi-squared distribution, the scaled distribution has an extra parameter ''τ''2, which scales the distribution horizontally and vertically, representing the inverse-variance of the original underlying process. Also, the scaled inverse chi-squared distribution is presented as the distribution for the inverse of the ''mean'' of ν squared deviat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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.B. Mandelbrot, The Pareto–Lévy Law and the Distribution of Income, International Economic Review 1960 https://www.jstor.org/stable/2525289 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 [...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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pearson Distribution
The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. History The Pearson system was originally devised in an effort to model visibly skewed observations. It was well known at the time how to adjust a theoretical model to fit the first two cumulants or moments of observed data: Any probability distribution can be extended straightforwardly to form a location-scale family. Except in pathological cases, a location-scale family can be made to fit the observed mean (first cumulant) and variance (second cumulant) arbitrarily well. However, it was not known how to construct probability distributions in which the skewness (standardized third cumulant) and kurtosis (standardized fourth cumulant) could be adjusted equally freely. This need became apparent when trying to fit known theoretical models to observed data that exhi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Gamma Distribution
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required. It is common among some Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution. Characteriza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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.B. Mandelbrot, The Pareto–Lévy Law and the Distribution of Income, International Economic Review 1960 https://www.jstor.org/stable/2525289 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 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levy0 LdistributionPDF
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 A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log–log Plot
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y=ax^k – appear as straight lines in a log–log graph, with the exponent corresponding to the slope, and the coefficient corresponding to the intercept. Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most commonly base 10 (common logs) are used. Relation with monomials Given a monomial equation y=ax^k, taking the logarithm of the equation (with any base) yields: \log y = k \log x + \log a. Setting X = \log x and Y = \log y, which corresponds to using a log–log graph, yields the equation: Y = mX + b where ''m'' = ''k'' is the slope of the line (gradient) and ''b'' = log ''a'' is the intercept on the (log ''y'')-axis, meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
