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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 Skellam distribution ： ウィキペディア英語版 Skellam distribution | kurtosis =$1/(\backslash mu\_1+\backslash mu\_2)\backslash ,$| entropy =| mgf =$e^+\backslash mu\_2e^\}$ }} The Skellam distribution is the discrete probability distribution of the difference $n\_1-n\_2$ of two statistically independent random variables $n\_1$ and $n\_2$ each having Poisson distributions with different expected values $\backslash mu\_1$ and $\backslash mu\_2$. It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in sports where all scored points are equal, such as baseball, hockey and soccer. The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application. The probability mass function for the Skellam distribution for a count difference $k=n\_1-n\_2$ from two Poisson-distributed variables with means $\backslash mu\_1$ and $\backslash mu\_2$ is given by: :$$ f(k;\mu_1,\mu_2)= e^ \left(\right)^I_(2\sqrt) where ''Ik''(''z'') is the modified Bessel function of the first kind. Note that since ''k'' is an integer we have that ''Ik''(''z'')=''I|k|''(''z''). == Derivation == Note that the probability mass function of a Poisson distribution for a count ''n'' with mean μ is given by :$$ f(n;\mu)=e^.\, for $n\; \backslash ge\; 0$ (and zero otherwise). The Skellam probability mass function for the difference of two counts $k=n\_1-n\_2$ is the cross-correlation of two Poisson distributions: (Skellam, 1946) :$$ f(k;\mu_1,\mu_2) =\sum_^\infty \!f(k\!+\!n;\mu_1)f(n;\mu_2) :$$ =e^\sum_^\infty Since the Poisson distribution is zero for negative values of the count , the second sum is only taken for those terms where $n\; >=\; 0$ and $n+k\; >=\; 0$. It can be shown that the above sum implies that :$\backslash frac=\backslash left(\backslash frac\backslash right)^k$ so that: :$$ f(k;\mu_1,\mu_2)= e^ \left(\right)^I_(2\sqrt) where ''I'' k(z) is the modified Bessel function of the first kind. The special case for $\backslash mu\_1=\backslash mu\_2(=\backslash mu)$ is given by Irwin (1937): :$$ f\left(k;\mu,\mu\right) = e^I_(2\mu). Note also that, using the limiting values of the modified Bessel function for small arguments, we can recover the Poisson distribution as a special case of the Skellam distribution for $\backslash mu\_2=0$. 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「Skellam 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.