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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 Beta prime distribution ： ウィキペディア英語版 Beta prime distribution \!| cdf =$I\_(\backslash alpha,\backslash beta)\; \}$ where $I\_x(\backslash alpha,\backslash beta)$ is the incomplete beta function| mean =$\backslash frac\; \backslash text\; \backslash beta>1$| median =| mode =$\backslash frac\; \backslash text\; \backslash alpha\backslash ge\; 1\backslash text\backslash !$| variance =$\backslash frac\; \backslash text\; \backslash beta>2$| skewness =$\backslash frac\backslash sqrt\}\; \backslash text\; \backslash beta>3$| kurtosis =| entropy =| mgf =| char =| }} In probability theory and statistics, the beta prime distribution (also known as inverted beta distribution or beta distribution of the second kind〔Johnson et al (1995), p248〕) is an absolutely continuous probability distribution defined for $x\; >\; 0$ with two parameters α and β, having the probability density function: : $f(x)\; =\; \backslash frac\}$ where ''B'' is a Beta function. The cumulative distribution function is :$F(x;\; \backslash alpha,\backslash beta)=I\_\}\backslash left\; (\backslash alpha,\; \backslash beta\; \backslash right)\; ,$ where ''I'' is the regularized incomplete beta function. The expectation value, variance, and other details of the distribution are given in the sidebox; for $\backslash beta>4$, the excess kurtosis is :$\backslash gamma\_2\; =\; 6\backslash frac$. While the related beta distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed as a probability, the beta prime distribution is the conjugate prior distribution of the parameter of a Bernoulli distribution expressed in odds. The distribution is a Pearson type VI distribution.〔 The mode of a variate ''X'' distributed as $\backslash beta^(\backslash alpha,\backslash beta)$ is $\backslash hat\; =\; \backslash frac$. Its mean is $\backslash frac$ if $\backslash beta>1$ (if $\backslash beta\; \backslash leq\; 1$ the mean is infinite, in other words it has no well defined mean) and its variance is $\backslash frac$ if $\backslash beta>2$. For :$E()=\backslash frac.$ For $k\backslash in\; \backslash mathbb$ with , this simplifies to :$E()=\backslash prod\_^\; \backslash frac.$ The cdf can also be written as :$\backslash frac\backslash !$ where $\_2F\_1$ is the Gauss's hypergeometric function 2F1 . Differential equation $$ \left\\right\} ==Generalization== Two more parameters can be added to form the generalized beta prime distribution. :$p\; >\; 0$ shape (real) $q\; >\; 0$ scale (real) having the probability density function: : $f(x;\backslash alpha,\backslash beta,p,q)\; =\; \backslash frac\}\backslash right)\}^\; \backslash left(\}\backslash right)\}^p\}\backslash right)^\}$ with mean : $\backslash frac)\backslash Gamma(\backslash beta-\backslash tfrac)\}\; \backslash text\; \backslash beta\; p>1$ and mode : $q\}\backslash right)\}^\backslash tfrac\; \backslash text\; \backslash alpha\; p\backslash ge\; 1\backslash !$ Note that if p=q=1 then the generalized beta prime distribution reduces to the standard beta prime distribution 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 prime 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.