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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Logarithmic distribution ： ウィキペディア英語版 Logarithmic distribution In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : $$ -\ln(1-p) = p + \frac + \frac + \cdots. From this we obtain the identity :$\backslash sum\_^\; \backslash frac\; \backslash ;\; \backslash frac\; =\; 1.$ This leads directly to the probability mass function of a Log(''p'')-distributed random variable: :$f(k)\; =\; \backslash frac\; \backslash ;\; \backslash frac$ for ''k'' ≥ 1, and where 0 < ''p'' < 1. Because of the identity above, the distribution is properly normalized. The cumulative distribution function is :$F(k)\; =\; 1\; +\; \backslash frac$ where ''B'' is the incomplete beta function. A Poisson compounded with Log(''p'')-distributed random variables has a negative binomial distribution. In other words, if ''N'' is a random variable with a Poisson distribution, and ''X''''i'', ''i'' = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(''p'') distribution, then :$\backslash sum\_^N\; X\_i$ has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution. R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance. The probability mass function ''ƒ'' of this distribution satisfies the recurrence relation : $f(k+1)\; =\; \backslash fracf(k);\; \backslash text\; f(1)\; =\; \backslash frac.$ ==See also== * Poisson distribution (also derived from a Maclaurin series) 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「Logarithmic 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.