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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 Bapat–Beg theorem ： ウィキペディア英語版 Bapat–Beg theorem In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables. Bapat and Beg published the theorem in 1989, though they did not offer a proof. A simple proof was offered by Hande in 1994. Often, all elements of the sample are obtained from the same population and thus have the same probability distribution. The Bapat–Beg theorem describes the order statistics when each element of the sample is obtained from a different statistical population and therefore has its own probability distribution.〔 ==Statement of theorem== Let $X\_1,X\_2,\backslash ldots,\; X\_n$ be independent real valued random variables with cumulative distribution functions respectively $F\_1(x),F\_2(x),\backslash ldots,F\_n(x)$. Write $X\_,X\_,\backslash ldots,\; X\_$ for the order statistics. Then the joint probability distribution of the $n\_1,n\_2,\backslash ldots,\; n\_k$ order statistics (with :$\backslash begin$ F_}(x_1,\ldots,x_k) & =\Pr ( X_\leq x_1 \and X_\leq x_2 \and\ldots\and X_ \leq x_k) \\ & =\sum_^n \ldots\sum_^\,\sum _^\frac, \end where : $P\_(x\_1,\backslash ldots,x\_k)\; =$ ::$$ \operatorname \begin F_1(x_1) \ldots F_1(x_1) & F_1(x_2)-F_1(x_1) \ldots F_1(x_2)-F_1(x_1) & \ldots & 1-F_1(x_k) \ldots 1-F_1(x_k) \\ F_2(x_1) \ldots F_2(x_1) & F_2(x_2)-F_2(x_1) \ldots F_2(x_2)-F_2(x_1) & \ldots & 1-F_2(x_k) \ldots 1-F_1(x_k )\\ \vdots & \vdots & & \vdots \\ \underbrace_ & \underbrace_ & \ldots & \underbrace_ \end is the permanent of the given block matrix. (The figures under the braces show the number of columns.)〔 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「Bapat–Beg theorem」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.