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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 Irwin-Hall distribution ： ウィキペディア英語版 Irwin–Hall distribution In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a uniform distribution.〔Johnson, N.L.; Kotz, S.; Balakrishnan, N. (1995) ''Continuous Univariate Distributions'', Volume 2, 2nd Edition, Wiley ISBN 0-471-58494-0(Section 26.9)〕 For this reason it is also known as the uniform sum distribution. The generation of pseudo-random numbers having an approximately normal distribution is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated. This distribution is sometimes confused with the Bates distribution, which is the mean (not sum) of ''n'' independent random variables uniformly distributed from 0 to 1. ==Definition== The Irwin–Hall distribution is the continuous probability distribution for the sum of ''n'' independent and identically distributed ''U''(0, 1) random variables: :$$ X = \sum_^n U_k. The probability density function (pdf) is given by :$$ f_X(x;n)=\frac\sum_^\left(-1\right)^k\left(x-k\right)^\sgn(x-k) where sgn(''x − k'') denotes the sign function: :$\backslash sgn\backslash left(x-k\backslash right)\; =\; \backslash begin$ -1 & x < k \\ 0 & x = k \\ 1 & x > k. \end Thus the pdf is a spline (piecewise polynomial function) of degree ''n'' − 1 over the knots 0, 1, ..., ''n''. In fact, for ''x'' between the knots located at ''k'' and ''k'' + 1, the pdf is equal to :$$ f_X(x;n) = \frac\sum_^ a_j(k,n) x^j where the coefficients ''aj(k,n)'' may be found from a recurrence relation over ''k'' :$$ a_j(k,n)=\begin 1&k=0, j=n-1\\ 0&k=0, ja_j(k-1,n) + \left(-1\right)^\choose j}k^ &k>0\end The coefficients are also (A188816 ) in OEIS. The coefficients for the cumulative distribution is (A188668 ). The mean and variance are ''n''/2 and ''n''/12, respectively. 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「Irwin–Hall 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.