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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 (a,b,0) class of distributions ： ウィキペディア英語版 (a,b,0) class of distributions In probability theory, the distribution of a discrete random variable $N$ is said to be a member of the (''a'', ''b'', 0) class of distributions if its probability mass function obeys : $\backslash frac,\; \backslash qquad\; k\; =\; 1,\; 2,\; 3,\; \backslash dots$ where $p\_k\; =\; P(N\; =\; k)$ (provided $a$ and $b$ exist and are real). There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF). More general distributions can be defined by fixing some initial values of ''pj'' and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of ''k'':〔 for example the logarithmic distribution and the discrete uniform distribution. The (''a'', ''b'', 0) class of distributions has important applications in actuarial science in the context of loss models.〔 == Properties == Sundt〔 proved that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to this class of distributions, with each distribution being represented by a different sign of ''a''. The more usual parameters of these distributions are determined by both ''a'' and ''b''. The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that $W\_N(x)\backslash ,$ denotes the probability generating function. p^k (1-p)^ |$\backslash frac$ |$\backslash frac$ |$(1-p)^n\backslash ,$ |$(px+(1-p))^\; \backslash ,$ |$np\backslash ,$ |$np(1-p)\; \backslash ,$ |- |Poisson |$e^\backslash frac\backslash ,$ |$0\backslash ,$ |$\backslash lambda\; \backslash ,$ |$e^\backslash ,$ |$e^\; \backslash ,$ |$\backslash lambda\backslash ,$ |$\backslash lambda\; \backslash ,$ |- |Negative binomial |$\backslash frac\backslash ,p^r\backslash ,(1-p)^k\; \backslash ,$ |$1-p\backslash ,$ |$(1-p)(r-1)\backslash ,$ |$p^r\; \backslash ,$ |$\backslash left(\; \backslash frac\backslash right)\; ^r\; \backslash ,$ |$\backslash frac\; \backslash ,$ |$\backslash frac\; \backslash ,$ |- |} 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「(a,b,0) class of distributions」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.