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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 Fisher's noncentral hypergeometric distribution ： ウィキペディア英語版 Fisher's noncentral hypergeometric distribution In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. It can also be defined as the conditional distribution of two or more binomially distributed variables dependent upon their fixed sum. The distribution may be illustrated by the following urn model. Assume, for example, that an urn contains ''m''1 red balls and ''m''2 white balls, totalling ''N'' = ''m''1 + ''m''2 balls. Each red ball has the weight ω1 and each white ball has the weight ω2. We will say that the odds ratio is ω = ω1 / ω2. Now we are taking balls randomly in such a way that the probability of taking a particular ball is proportional to its weight, but independent of what happens to the other balls. The number of balls taken of a particular color follows the binomial distribution. If the total number ''n'' of balls taken is known then the conditional distribution of the number of taken red balls for given ''n'' is Fisher's noncentral hypergeometric distribution. To generate this distribution experimentally, we have to repeat the experiment until it happens to give ''n'' balls. If we want to fix the value of ''n'' prior to the experiment then we have to take the balls one by one until we have ''n'' balls. The balls are therefore no longer independent. This gives a slightly different distribution known as Wallenius' noncentral hypergeometric distribution. It is far from obvious why these two distributions are different. See the entry for noncentral hypergeometric distributions for an explanation of the difference between these two distributions and a discussion of which distribution to use in various situations. The two distributions are both equal to the (central) hypergeometric distribution when the odds ratio is 1. Unfortunately, both distributions are known in the literature as "the" noncentral hypergeometric distribution. It is important to be specific about which distribution is meant when using this name. Fisher's noncentral hypergeometric distribution was first given the name extended hypergeometric distribution (Harkness, 1965), and some authors still use this name today. == Univariate distribution == \right\rfloor \, , where $A=\backslash omega-1$, $B\; =\; m\_1\; +\; n\; -\; N\; -(m\_1+n+2)\backslash omega$, $C\; =\; (m\_1+1)(n+1)\backslash omega$.| variance =$\backslash frac\; -\; \backslash left(\; \backslash frac\; \backslash right)^2$, where ''P''''k'' is given above.| skewness =| kurtosis =| entropy =| mgf =| char = }} The probability function, mean and variance are given in the table to the right. An alternative expression of the distribution has both the number of balls taken of each color and the number of balls not taken as random variables, whereby the expression for the probability becomes symmetric. The calculation time for the probability function can be high when the sum in ''P''0 has many terms. The calculation time can be reduced by calculating the terms in the sum recursively relative to the term for ''y'' = ''x'' and ignoring negligible terms in the tails (Liao and Rosen, 2001). The mean can be approximated by: :$\backslash mu\; \backslash approx\; \backslash frac\; \backslash bigg/\; \backslash left(\; \backslash frac+\; \backslash frac+\; \backslash frac+\; \backslash frac\; \backslash right)$ . Better approximations to the mean and variance are given by Levin (1984, 1990), McCullagh and Nelder (1989), Liao (1992), and Eisinga and Pelzer (2011). The saddlepoint methods to approximate the mean and the variance suggested Eisinga and Pelzer (2011) offer extremely accurate results. 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「Fisher's noncentral hypergeometric 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.