翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク beta distribution ： ウィキペディア英語版 beta distribution \!| cdf =$I\_x(\backslash alpha,\backslash beta)\backslash !$| mean =$\backslash operatorname()\; =\; \backslash frac\backslash !$ $\backslash operatorname(X)\; =\; \backslash psi(\backslash alpha)\; -\; \backslash psi(\backslash alpha\; +\; \backslash beta)\backslash !$ (see digamma function and see section: Geometric mean)| median =$\backslash beginI\_\}^(\backslash alpha,\backslash beta)\backslash text\backslash \backslash ()$ \approx \frac } }\text\alpha, \beta >1\end| mode =$\backslash frac\backslash !$ for α, β >1| variance =$\backslash operatorname()\; =\; \backslash frac\backslash !$ $\backslash operatorname(X)\; =\; \backslash psi\_1(\backslash alpha)\; -\; \backslash psi\_1(\backslash alpha\; +\; \backslash beta)\backslash !$ (see trigamma function and see section: Geometric variance)| skewness =$\backslash frac\}$| kurtosis =$\backslash frac$| entropy =$\backslash begin\backslash ln\backslash Beta(\backslash alpha,\backslash beta)-(\backslash alpha-1)\backslash psi(\backslash alpha)-(\backslash beta-1)\backslash psi(\backslash beta)\backslash \backslash ()$ +(\alpha+\beta-2)\psi(\alpha+\beta)\end| mgf =$1\; +\backslash sum\_^\; \backslash left(\; \backslash prod\_^\; \backslash frac\; \backslash right)\; \backslash frac$| char = see section: Fisher information matrix }} In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval (1 ) parametrized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. For example, it has been used as a statistical description of allele frequencies in population genetics; time allocation in project management / control systems;〔 sunshine data; variability of soil properties; proportions of the minerals in rocks in stratigraphy; and heterogeneity in the probability of HIV transmission. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. For example, the beta distribution can be used in Bayesian analysis to describe initial knowledge concerning probability of success such as the probability that a space vehicle will successfully complete a specified mission. The beta distribution is a suitable model for the random behavior of percentages and proportions. The usual formulation of the beta distribution is also known as the beta distribution of the first kind, whereas ''beta distribution of the second kind'' is an alternative name for the beta prime distribution. ==History== The first systematic modern discussion of the beta distribution is probably due to Karl Pearson FRS (27 March 1857 – 27 April 1936), an influential English mathematician who has been credited with establishing the discipline of mathematical statistics. In Pearson's papers〔〔 the beta distribution is couched as a solution of a differential equation: Pearson's Type I distribution which it is essentially identical to except for arbitrary shifting and re-scaling (the beta and Pearson Type I distributions can always be equalized by proper choice of parameters). In fact, in several English books and journal articles in the few decades prior to World War II, it was common to refer to the beta distribution as Pearson's Type I distribution. William P. Elderton (1877–1962) in his 1906 monograph "Frequency curves and correlation" further analyzes the beta distribution as Pearson's Type I distribution, including a full discussion of the method of moments for the four parameter case, and diagrams of (what Elderton describes as) U-shaped, J-shaped, twisted J-shaped, "cocked-hat" shapes, horizontal and angled straight-line cases. Elderton wrote "I am chiefly indebted to Professor Pearson, but the indebtedness is of a kind for which it is impossible to offer formal thanks." Elderton in his 1906 monograph 〔 provides an impressive amount of information on the beta distribution, including equations for the origin of the distribution chosen to be the mode, as well as for other Pearson distributions: types I through VII. Elderton also included a number of appendixes, including one appendix ("II") on the beta and gamma functions. In later editions, Elderton added equations for the origin of the distribution chosen to be the mean, and analysis of Pearson distributions VIII through XII. As remarked by Bowman and Shenton 〔 "Fisher and Pearson had a difference of opinion in the approach to (parameter) estimation, in particular relating to (Pearson's method of) moments and (Fisher's method of) maximum likelihood in the case of the Beta distribution." Also according to Bowman and Shenton, "the case of a Type I (beta distribution) model being the center of the controversy was pure serendipity. A more difficult model of 4 parameters would have been hard to find." Ronald Fisher (17 February 1890 – 29 July 1962) was one of the giants of statistics in the first half of the 20th century, and his long running public conflict with Karl Pearson can be followed in a number of articles in prestigious journals. For example, concerning the estimation of the four parameters for the beta distribution, and Fisher's criticism of Pearson's method of moments as being arbitrary, see Pearson's article "Method of moments and method of maximum likelihood" (published three years after his retirement from University College, London, where his position had been divided between Fisher and Pearson's son Egon) in which Pearson writes "I read (Koshai's paper in the Journal of the Royal Statistical Society, 1933) which as far as I am aware is the only case at present published of the application of Professor Fisher's method. To my astonishment that method depends on first working out the constants of the frequency curve by the (Pearson) Method of Moments and then superposing on it, by what Fisher terms "the Method of Maximum Likelihood" a further approximation to obtain, what he holds, he will thus get, "more efficient values" of the curve constants." David and Edwards's treatise on the history of statistics cites the first modern treatment of the beta distribution, in 1911, using the beta designation that has become standard, due to Corrado Gini,(May 23, 1884 – March 13, 1965), an Italian statistician, demographer, and sociologist, who developed the Gini coefficient. N.L.Johnson and S.Kotz, in their comprehensive and very informative monograph on leading historical personalities in statistical sciences credit Corrado Gini as "an early Bayesian...who dealt with the problem of eliciting the parameters of an initial Beta distribution, by singling out techniques which anticipated the advent of the so called empirical Bayes approach." Bayes, in a posthumous paper published in 1763 by Richard Price, obtained a beta distribution as the density of the probability of success in Bernoulli trials (see the section titled "Applications, Bayesian inference" in this article), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「beta distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.