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第1,953行： − + − Computing <math>P(k;\lambda)</math> for given <math>k</math> and <math>\lambda</math> is a trivial task that can be accomplished by using the standard definition of <math>P(k;\lambda)</math> in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λ<sup>''k''</sup> and ''k''!. The fraction of λ<sup>''k''</sup> to ''k''! can also produce a rounding error that is very large compared to ''e''<sup>−λ</sup>, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as − Computing <math>P(k;\lambda)</math> for given <math>k</math> and <math>\lambda</math> is a trivial task that can be accomplished by using the standard definition of <math>P(k;\lambda)</math> in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: λ<sup>k</sup> and k!. The fraction of λ<sup>k</sup> to k! can also produce a rounding error that is very large compared to e<sup>−λ</sup>, and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as+ − − 计算 < math > p (k; lambda) </math > 对于给定的 < math > k </math > 和 < math > lambda </math > 是一项琐碎的任务，可以通过使用 < math > p (k; lambda) </math > 的标准定义来完成，包括指数函数、幂函数和阶乘函数。然而，传统上对泊松分布的定义包含了两个容易在计算机上溢出的术语: < sup > k </sup > 和 k。分数 < sup > k </sup > 到 k！也可能产生舍入误差，与 e < sup >-</sup > 相比，舍入误差非常大，因此给出错误的结果。因此，对于数值稳定性来说，泊松概率质量函数应该被评估为 − --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布”改为“泊松分布” − <math>\!f(k; \lambda)= \exp \left[ k\ln \lambda - \lambda - \ln \Gamma (k+1) \right],</math> − − = exp left [ k ln lambda-lambda-ln Gamma (k + 1) right ] ，</math > − − which is mathematically equivalent but numerically stable. The natural logarithm of the [[Gamma function]] can be obtained using the <code>lgamma</code> function in the [[C (programming language)|C]] standard library (C99 version) or [[R (programming language)|R]], the <code>gammaln</code> function in [[MATLAB]] or [[SciPy]], or the <code>log_gamma</code> function in [[Fortran]] 2008 and later. − − which is mathematically equivalent but numerically stable. The natural logarithm of the Gamma function can be obtained using the <code>lgamma</code> function in the C standard library (C99 version) or R, the <code>gammaln</code> function in MATLAB or SciPy, or the <code>log_gamma</code> function in Fortran 2008 and later. − − 这在数学上是等价的，但在数值上是稳定的。函数的自然对数可以使用 c 标准库(C99版本)中的 < code > lgamma </code > 函数或 r，MATLAB 或 SciPy 中的 < code > gammaln </code > 函数，或 Fortran 2008及更高版本中的 < code > log _ Gamma </code > 函数来获得。 + − Some computing languages provide built-in functions to evaluate the Poisson distribution, namely+ − Some computing languages provide built-in functions to evaluate the Poisson distribution, namely+ − 一些计算语言提供了''' 内置函数Built-in functions'''来评估泊松分布+ − --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布”改为“泊松分布” − − + − * [[Mathematica]]: univariate Poisson distribution as <code>PoissonDistribution[<math>\lambda</math>]</code>,<ref name="WLPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/PoissonDistribution.html |title = Wolfram Language: PoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref> bivariate Poisson distribution as <code>MultivariatePoissonDistribution[<math>\theta_{12}</math>,{ <math>\theta_1 - \theta_{12}</math>, <math>\theta_2 - \theta_{12}</math>}]</code>,.<ref name="WLMvPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html |title = Wolfram Language: MultivariatePoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref>+
→Evaluating the Poisson distribution 计算泊松分布
=== ''' Evaluating the Poisson distribution 计算泊松分布'''===
===计算泊松分布===
计算<math>P(k;\lambda)</math>对于给定的<math>k</math>和<math>\lambda</math>是一项琐碎的任务，可以通过使用<math>P(k;\lambda)</math>的标准定义来完成，包括指数函数、幂函数和阶乘函数。然而，传统上对泊松分布的定义包含了两个容易在计算机上溢出的术语: λ<sup>''k''</sup>!和 ''k''。分数λ<sup>''k''</sup> 到''k''!也可能产生舍入误差，与''e''<sup>−λ</sup>相比，舍入误差非常大，因此给出错误的结果。因此，对于数值稳定性来说，泊松概率质量函数应该被评估为
:<math>\!f(k; \lambda)= \exp \left[ k\ln \lambda - \lambda - \ln \Gamma (k+1) \right],</math>
:<math>\!f(k; \lambda)= \exp \left[ k\ln \lambda - \lambda - \ln \Gamma (k+1) \right],</math>
这在数学上是等价的，但在数值上是稳定的。函数的自然对数可以使用 c 标准库(C99版本)中的<code>lgamma</code>函数或 R，MATLAB 或 SciPy 中的 <code>gammaln</code>函数，或 Fortran 2008及更高版本中的<code>log_gamma</code>函数来获得。
一些计算语言提供了''' 内置函数来计算泊松分布
* R语言：function <code>dpois(x, lambda)</code>;
* Microsoft Excel：function <code>POISSON( x, mean, cumulative)</code>, with a flag to specify the cumulative distribution;
* [[R (programming language)|R]]: function <code>dpois(x, lambda)</code>;
* [[Microsoft Excel|Excel]]: function <code>POISSON( x, mean, cumulative)</code>, with a flag to specify the cumulative distribution;
* Mathematica：单变量泊松分布为<code>PoissonDistribution[<math>\lambda</math>]</code>,<ref name="WLPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/PoissonDistribution.html |title = Wolfram Language: PoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref> ，二元泊松分布为<code>MultivariatePoissonDistribution[<math>\theta_{12}</math>,{ <math>\theta_1 - \theta_{12}</math>, <math>\theta_2 - \theta_{12}</math>}]</code>.<ref name="WLMvPoissonRefPage">{{cite web |url = http://reference.wolfram.com/language/ref/MultivariatePoissonDistribution.html |title = Wolfram Language: MultivariatePoissonDistribution reference page |website = wolfram.com |access-date = 2016-04-08 }}</ref>
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===从泊松分布中抽取随机量===
===从泊松分布中抽取随机量===