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第2行： − {{short description|Discrete probability distribution}} − {{Infobox probability distribution − | name = Poisson Distribution − | type = mass − | pdf_image = [[File:poisson pmf.svg|325px]] − | pdf_caption = The horizontal axis is the index ''k'', the number of occurrences. ''λ'' is the expected rate of occurrences. The vertical axis is the probability of ''k'' occurrences given ''λ''. The function is defined only at integer values of ''k''; the connecting lines are only guides for the eye. − | cdf_image = [[File:poisson cdf.svg|325px]] − | cdf_caption = The horizontal axis is the index ''k'', the number of occurrences. The CDF is discontinuous at the integers of ''k'' and flat everywhere else because a variable that is Poisson distributed takes on only integer values. − | notation = <math>\operatorname{Pois}(\lambda)</math> − | parameters = <math>\lambda\in (0, \infty) </math> (rate) − | support = <math>k \in \mathbb{N}_0</math> ([[Natural numbers]] starting from 0) − | pdf = <math>\frac{\lambda^k e^{-\lambda}}{k!}</math> − | cdf = <math>\frac{\Gamma(\lfloor k+1\rfloor, \lambda)}{\lfloor k\rfloor !}</math>, or <math>e^{-\lambda} \sum_{i=0}^{\lfloor k\rfloor} \frac{\lambda^i}{i!}\ </math>, or <math>Q(\lfloor k+1\rfloor,\lambda)</math> − (for <math>k\ge 0</math>, where <math>\Gamma(x, y)</math> is the [[upper incomplete gamma function]], <math>\lfloor k\rfloor</math> is the [[floor function]], and Q is the [[regularized gamma function]]) − | mean = <math>\lambda</math> − | median = <math>\approx\lfloor\lambda+1/3-0.02/\lambda\rfloor</math> − | mode = <math>\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor</math> − | variance = <math>\lambda</math> − | skewness = <math>\lambda^{-1/2}</math> − | kurtosis = <math>\lambda^{-1}</math> − | entropy = <math>\lambda[1 - \log(\lambda)] + e^{-\lambda}\sum_{k=0}^\infty \frac{\lambda^k\log(k!)}{k!}</math> − (for large <math>\lambda</math>) − <math>\frac{1}{2}\log(2 \pi e \lambda) - \frac{1}{12 \lambda} - \frac{1}{24 \lambda^2} -{}</math><br><math>\qquad \frac{19}{360 \lambda^3} + O\left(\frac{1}{\lambda^4}\right)</math><!--formula split with \qquad indent--> − | pgf = <math>\exp[\lambda(z - 1)]</math> − | mgf = <math>\exp[\lambda (e^{t} - 1)]</math> − | char = <math>\exp[\lambda (e^{it} - 1)]</math> − | fisher = <math>\frac{1}{\lambda}</math> − }} 第35行：
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第1,100行： − 一个简单的方法来产生一个二变量的泊松分佈分布: 取3个独立的 Poisson 分布 < math > y _ 1，y _ 2，y _ 3 </math > ，用 < math > lambda _ 1，lambda _ 2，lambda _ 3 </math > 然后设置 < math > x _ 1 = y _ 1 + y _ 3，x _ 2 = y _ 2 + y _ 3 </math > 。二元概率密度函数变量的泊松分佈变量是+ − + 第1,168行：
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第1,662行： − 单一额外观察的后验预测分布是负二项分布，有时称为泊松分佈。+ − + 第1,728行：
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第1,727行： − 泊松分佈过程与泊松过程有关。它适用于各种离散性质的现象(也就是说，那些可能发生0,1,2,3，... 在给定时间内或在给定区域) ，只要现象发生的概率在时间或空间上是常数。可以被模仿为泊松分佈的活动包括:+ − + 第1,822行：
第1,794行： − 泊松分佈(黑线)与[二项分布(红圈) ，n = 20(蓝圈) ，n = 1000(绿圈)的比较。所有分布的平均值都是5。水平轴显示事件的数量 k。随着 n 变得越来越大，泊松分佈变成了一个越来越好的平均二项分布。]+ − + 第1,829行：
第1,801行： − + − + − 在这种情况下，二项分布收敛于泊松极限定理所称的泊松分佈。+ − + 第1,859行：
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第1,855行： − 法律一词有时被用作概率分布的同义词，法律的趋同意味着分配的趋同。因此，泊松分佈有时被称为“小数定律” ，因为它是一个事件发生次数的概率分布，这个事件很少发生，但却有很多机会发生。小数定律》是拉迪斯劳斯·博特基威茨的一本关于泊松分佈的书，出版于1898年。+ − + − + 第1,899行：
第1,871行： − 泊松分佈是位于某个有限区域的泊松过程的点数。更具体地说，如果 d 是某个区域空间，例如欧几里德空间 r < sup > d </sup > ，对于这个区域 | d | ，区域的面积、体积或者更一般地说，区域的勒贝格测度是有限的，如果表示 d 中的点数，那么+ − + 第1,911行：
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第1,948行： − 泊松分佈为专用软件库提出了两个不同的任务: 评估分布 < math > p (k; lambda) </math > ，并根据分布绘制随机数。+ − + 第1,986行：
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第2,010行： − + + + + + + + + + + + − A simple algorithm to generate random Poisson-distributed numbers ([[pseudo-random number sampling]]) has been given by [[Donald Knuth|Knuth]]:{{r|Knuth1997|p=137-138}} − A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth: − 给出了一个产生随机泊松分布数(伪随机数抽样)的简单算法:+ + + + + − + − + − 算法泊松随机数(Knuth) :+ 第2,062行：
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第2,064行： − + − + − + − '''while''' p > L.+ − while p > L.+ − 而 p > l。+ − '''return''' k − 1.+ − return k − 1.+ − return k − 1.+ + + − The complexity is linear in the returned value ''k'', which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see {{slink||References}} below.+ − The complexity is linear in the returned value k, which is λ on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see below.+ − 返回值 k 的复杂度是线性的，平均为。还有许多其他的算法可以改进这一点。一些是在 Ahrens & Dieter，见下面。+ + + − For large values of λ, the value of L = ''e''<sup>−λ</sup> may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that ''e''<sup>−STEP</sup> does not underflow: {{Citation needed|reason=Original source is missing|date=March 2019}}+ − For large values of λ, the value of L = e<sup>−λ</sup> may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e<sup>−STEP</sup> does not underflow: + − 对于较大的值，l = e </sup >-</sup > 的值可能非常小，以至于很难表示。这可以通过改变算法来解决，该算法使用附加参数 STEP，使得 e < sup >-STEP </sup > 不会底流:+ − --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“底流”改为“下溢”+ − '''algorithm''' ''poisson random number (Junhao, based on Knuth)'':+ − algorithm poisson random number (Junhao, based on Knuth):+ − 泊松随机数算法(Junhao，基于 Knuth) :+ − '''init''':+ − init: + − 初始化:+ − '''Let''' λLeft ← λ, k ← 0 and p ← 1.+ − Let λLeft ← λ, k ← 0 and p ← 1.+ − Let λLeft ← λ, k ← 0 and p ← 1.+ − '''do''': + − do: − 做: − k ← k + 1.+ − k ← k + 1.+ − k ← k + 1.+ − Generate uniform random number u in (0,1) and '''let''' p ← p × u. − Generate uniform random number u in (0,1) and let p ← p × u. − 在(0,1)中生成均匀随机数 u 并且设 p ← p u。+ − '''while''' p < 1 and λLeft > 0:+ − while p < 1 and λLeft > 0:+ − while p < 1 and λLeft > 0: − '''if''' λLeft > STEP: − if λLeft > STEP:+ − if λLeft > STEP:+ − p ← p × ''e''<sup>STEP</sup>+ − p ← p × e<sup>STEP</sup> − p ← p × e<sup>STEP</sup> − λLeft ← λLeft − STEP+ − λLeft ← λLeft − STEP+ − λLeft ← λLeft − STEP+ − '''else''':+ − else:+ − 其他:+ − p ← p × ''e''<sup>λLeft</sup>+ − p ← p × e<sup>λLeft</sup>+ − p ← p × e<sup>λLeft</sup>+ − λLeft ← 0+ − λLeft ← 0+ − λLeft ← 0+ − + − + − 同时 p > 1。+ − '''return''' k − 1.+ − return k − 1.+ − return k − 1.+ + + − The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near ''e''<sup>700</sup>, so 500 shall be a safe ''STEP''.+ − The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near e<sup>700</sup>, so 500 shall be a safe STEP.+ − STEP 的选择取决于溢出'''<font color="#ff8000"> 阈值Threshold</font>'''。对于双精度浮点格式，'''<font color="#ff8000"> 阈值Threshold</font>'''接近 e < sup > 700 </sup > ，因此500应该是一个安全的 STEP。+ + + − Other solutions for large values of λ include [[rejection sampling]] and using Gaussian approximation.+ − Other solutions for large values of λ include rejection sampling and using Gaussian approximation.+ − 其他λ的大数值的求解包括抑制取样和使用'''<font color="#ff8000"> 高斯近似Gaussian approximation</font>'''。 + + − [[Inverse transform sampling]] is simple and efficient for small values of λ, and requires only one uniform random number ''u'' per sample. Cumulative probabilities are examined in turn until one exceeds ''u''. − Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.+ − 逆变换采样对于λ小数值的样本是简单有效的，并且每个样本只需要一个均匀一致的随机数 u。依次检查累积概率，直到其超过 u。 + − '''algorithm''' ''Poisson generator based upon the inversion by sequential search'':{{r|Devroye1986|p=505}}+ − + − algorithm Poisson generator based upon the inversion by sequential search:+ − + − 基于顺序检索反演的'''<font color="#ff8000"> 泊松发生器算法Algorithm Poisson generator </font>''':+ − + − '''init''':+ − + − init:+ − + − 初始化:+ − + − '''Let''' x ← 0, p ← ''e''<sup>−λ</sup>, s ← p.+ − + − Let x ← 0, p ← e<sup>−λ</sup>, s ← p.+ − + − Let x ← 0, p ← e<sup>−λ</sup>, s ← p.+ − + − Generate uniform random number u in [0,1].+ − + − Generate uniform random number u in [0,1].+ − + − 在[0,1]中生成均匀随机数 u。+ − + − '''while''' u > s '''do''':+ − + − while u > s do:+ − + − 做以下步骤:+ − + − x ← x + 1.+ − + − x ← x + 1.+ − + − x ← x + 1.+ − + − p ← p × λ / x.+ − + − p ← p × λ / x.+ − + − p ← p × λ / x.+ − + − s ← s + p.+ − + − s ← s + p.+ − + − s ← s + p.+ − + − '''return''' x.+ − + − return x.+ − + − 返回 x。+ − + − + − + − + − + − The distribution was first introduced by [[Siméon Denis Poisson]] (1781–1840) and published together with his probability theory in his work ''Recherches sur la probabilité des jugements en matière criminelle et en matière civile''(1837).{{r|Poisson1837|p=205-207}} The work theorized about the number of wrongful convictions in a given country by focusing on certain [[random variable]]s ''N'' that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a [[time]]-interval of given length. The result had already been given in 1711 by [[Abraham de Moivre]] in ''De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus'' .{{r|deMoivre1711|p=219}}{{r|deMoivre1718|p=14-15}}{{r|deMoivre1721|p=193}}{{r|Johnson2005|p=157}} This makes it an example of [[Stigler's law]] and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.{{r|Stigler1982|Hald1984}}+ − + − The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile(1837). The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.+ − + − The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile(1837).+ − 这种分布最早由西蒙·丹尼斯·泊松（1781-1840）提出，与他的概率理论一起发表在他的著作《苏拉河畔的调查》中，'''<font color="#32CD32">sur la probabilité des jugements en matière criminelle et en matière civile(1837)</font>'''+ − 这项工作通过关注某些随机变量 n (其中包括在给定时间间隔内发生的离散事件(有时称为“事件”或“到达事件”)的数量)来推断某一国家的错误定罪数量。这个结果早在1711年就已经在《亚伯拉罕·棣莫弗》给出了。在 Ludis 举行的猜测活动。这使它成为斯蒂格勒定律的一个例子，也使一些作者提出，'''<font color="#ff8000"> 泊松分佈Poisson distribution </font>'''应该以德莫伊弗雷的名字命名。+ − + − --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”+ − + − In 1860, [[Simon Newcomb]] fitted the Poisson distribution to the number of stars found in a unit of space.{{r|Newcomb1860}}+ − + − In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.+ − + − 1860年，Simon Newcomb 将'''<font color="#ff8000"> 泊松分佈Poisson distribution </font>'''与天文台一个空间单位中发现的恒星数量进行了比较。+ − + − A further practical application of this distribution was made by [[Ladislaus Bortkiewicz]] in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks;{{r|vonBortkiewitsch1898|p=23-25}} this experiment introduced the Poisson distribution to the field of [[reliability engineering]].+ − + − A further practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks; this experiment introduced the Poisson distribution to the field of reliability engineering.+ − + − 这种分布的进一步实际应用是在1898年，当时拉迪斯劳斯·博特基威茨被赋予任务调查普鲁士军队中被马踢意外杀死的士兵人数; 这个实验将'''<font color="#ff8000"> 泊松分佈Poisson distribution </font>'''引入可靠性技术领域。+ − + − --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“泊松分布”+ − + − + − + − {{div col |colwidth = 18em }} − − * [[Compound Poisson distribution]] − − * [[Conway–Maxwell–Poisson distribution]] − − * [[Erlang distribution]] − − * [[Hermite distribution]] − − * [[Index of dispersion]] − − * [[Negative binomial distribution]] − − * [[Poisson clumping]] − − * [[Poisson point process]] − − * [[Poisson regression]] − − * [[Poisson sampling]] − − * [[Poisson wavelet]] − − * [[Queueing theory]] − − * [[Renewal theory]] − − * [[Robbins lemma]] − − * [[Skellam distribution]] − − * [[Tweedie distribution]] − − * [[Zero-inflated model]] − − * [[Zero-truncated Poisson distribution]] − − − − − − − − − − {{Reflist − − − − {通货再膨胀 − − − − − − 2012年10月15日 − − <ref name=Haight1967> − − − − − − + + + + + + + + + + + + + + + + + + + + + + + + + − + + + + + + + + + + + + + + + + + + − + − + − + − + − + − 1 = Haight+ − + − + − + − + − + − 1 = Frank a.+ − + − + − + − + − + − + − + − + − + − + − + − 2012年3月24日 | publisher = 约翰威立+ − + − + − + − + − + − | 地点: 美国纽约州纽约 − − − − − − 1967年 − − |isbn=978-0-471-33932-8 − − |isbn=978-0-471-33932-8 − − − − − − }}</ref> − − } </ref > − − − − <ref name=Brooks2007> − − < ref name = brooks2007 > − − + + + + + + + + − + − { citation+ − + − + − + − |last=Brooks |first=E. Bruce+ − + − | last = Brooks | first = e.布鲁斯 − − − − − − 2012年10月14日 | 标题: 统计数据 & # 124; 泊松分佈 − − − − |series=Warring States Project |publisher=Umass.edu − − 战国计划 | 出版社 = Umass.edu − − |date=2007-08-24 − − |date=2007-08-24 − − | date = 2007-08-24 − − − − |accessdate=2014-04-18 − − 2014-04-18 − − |url=https://www.umass.edu/wsp/resources/reference/poisson/ − − |url=https://www.umass.edu/wsp/resources/reference/poisson/ − − Https://www.umass.edu/wsp/resources/reference/poisson/ − − − − }}</ref> − − } </ref > − − − − <ref name=Poisson1837> − − < ref name = poisson1837 > − − + − {{citation+ − + − { citation+ − + − + − + − + − + − 1 = Poisson+ − + − + − + − + − + − 1 = Siméon d.+ − + − + − + − |title=Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés + − + − + − + − |trans-title=Research on the Probability of Judgments in Criminal and Civil Matters+ − + − + − − − − − − |language=French − − 语言 = 法语 − − |publisher=Bachelier − − − − − − − − − − | 地点: 法国巴黎 − − − − − − 1837年 − − − − − − Https://gallica.bnf.fr/ark:/12148/bpt6k110193z/f218.image − − + − }}</ref>+ − + − } </ref >+ − + − <ref name=Koehrsen2019>+ − + − + − + − + − + + + + + − + + + + + + + + + − + − { citation+ − + − + − + − |last1=Koehrsen+ − + − 1 = Koehrsen+ − + − + − + − |first1=William+ − + − 1 = William+ − + − − − |title=The Poisson Distribution and Poisson Process Explained − − 泊松分佈和泊松过程的解释 − − − − − − 走向数据科学 − − − − − − | date = 2019-01-20 − − − − − − Https://towardsdatascience.com/the-poisson-distribution-and-poisson-process-explained-4e2cb17d459 − − − − − − 2019-09-19 − − + + + + + + + + + − + − } </ref >+ − + − <ref name="Ugarte2016">+ − + − + − + − + − + − + − + − {{Citation+ − + − {引文+ − + − + − + − + − + − 1 = Ugarte | first1 = Maria Dolores+ − + − + − + − + − + − 2 = Militino | first2 = Ana f.+ − + − + − + − + − + − 3 = Arnholt | first3 = Alan t.+ − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 2016年+ − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − | isbn = 978-1-4665-0439-4+ − + − + − + − }}+ − + − }}+ − − − − − − </ref > − − <ref name=deMoivre1711> − − − − − − + + + + + + + + + − + − { citation+ − + − + − + − + − + − 1 = de Moivre | first1 = Abraham+ − + − |title=De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus+ − + − + − + − + − + − |trans-title=On the Measurement of Chance, or, on the Probability of Events in Games Depending Upon Fortuitous Chance+ − + − |trans-title=On the Measurement of Chance, or, on the Probability of Events in Games Depending Upon Fortuitous Chance + − + − | trans-title = 关于运气的测量，或者说，关于依靠偶然机会的比赛事件的概率+ − + − + − + − |language=Latin+ − + − 语言 = 拉丁语+ − + − + − − − − 英国皇家学会哲学学报 − − − − |year=1711 |volume=27 |issue=329 |pages=213–264 − − 1711 | volume = 27 | issue = 329 | pages = 213-264 − − − − − − − − − − − − 免费访问 − − + + + + + + + + − + − } </ref >+ − + − <ref name=deMoivre1718>+ − + − + − + − < ref name = demoivre1718 > + − + − + − + − {{citation+ − + − { citation+ − + − + − + − + − + − 1 = de Moivre | first1 = Abraham+ − + − + − + − + − + − 1718年+ − + − + − + − + − + − − − − − |publisher=W. Pearson − − − − − − − − − − − − − − Https://books.google.com/books?id=3epac6qpbumc&pg=pa14 − − + − }}</ref>+ − + − } </ref >+ − + − <ref name=deMoivre1721>+ − + − + − + − + − − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − + − { citation+ − + − + − + − + − + − 1 = de Moivre | first1 = Abraham+ − + − + − + − + − + − 1721年+ − + − + − + − + − + − 第二章: 运气法则+ − + − + − + − + − + − | title = 从 MDCC (洛索普先生结束的地方)到 MDCCXX 年的哲学事务。在将军的领导下，删节和修改+ − + − + − + − + − + − + − + − + − + − + − + − 语言 = 拉丁语+ − + − + − + − + − − 1 = Motte | editor-first1 = Benjamin − − − − − − − − − − − − | 地点: 英国伦敦 − − − − − − − − + − }}</ref>+ − + − } </ref >+ − + − <ref name="Johnson2005">+ − + − + − + − + − + + + − + + + + + + + + + − + − { citation+ − + − + − + − + − + − 1 = Johnson | first1 = Norman l.+ − + − + − + − + − + − 2 = Kemp | first2 = Adrienne w.+ − + − + − + − + − + − 3 = Kotz | first3 = Samuel+ − + − − − |chapter=Poisson Distribution − − | chapter = 泊松分佈 − − − − − − − − − − − − − − |edition=3rd − − |edition=3rd − − 3 rd − − |publisher=John Wiley & Sons, Inc. − − |publisher=John Wiley & Sons, Inc. − − 2012年10月21日 | 出版商 = 约翰威立。 − − |location=New York, NY, USA − − |location=New York, NY, USA − − | 地点: 美国纽约州纽约 − − |isbn=978-0-471-27246-5 − − |isbn=978-0-471-27246-5 − − 978-0-471-27246-5 − − |doi=10.1002/0471715816 − − |doi=10.1002/0471715816 − − 10.1002/0471715816 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Helske2017> − − <ref name=Helske2017> − − < ref name = helske2017 > − − {{cite arxiv − − {{cite arxiv − − {{cite arxiv − − |last=Helske |first=Jouni − − |last=Helske |first=Jouni − − 最后 = Helske | first = Jouni − − |date=2017 − − |date=2017 − − 2017年 − − |title=KFAS: Exponential family state space models in R − − |title=KFAS: Exponential family state space models in R − − | title = KFAS: r 的指数族状态空间模型 − − |eprint=1612.01907 − − |eprint=1612.01907 − − 1612.01907 − − |class=stat.CO − − |class=stat.CO − − | class = stat.CO − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Stigler1982> − − <ref name=Stigler1982> − − < ref name = stigler1982 > − − {{citation − − {{citation − − { citation − − |last1=Stigler |first1=Stephen M. − − |last1=Stigler |first1=Stephen M. − − 1 = Stigler | first1 = Stephen m. − − |title=Poisson on the Poisson Distribution − − |title=Poisson on the Poisson Distribution − − 泊松分佈上的泊松 − − |journal=Statistics & Probability Letters − − |journal=Statistics & Probability Letters − − | journal = Statistics & Probability Letters − − |year=1982 |volume=1 |issue=1 |pages=33–35 − − |year=1982 |volume=1 |issue=1 |pages=33–35 − − 1982 | volume = 1 | issue = 1 | pages = 33-35 − − |doi=10.1016/0167-7152(82)90010-4 − − |doi=10.1016/0167-7152(82)90010-4 − − | doi = 10.1016/0167-7152(82)90010-4 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Hald1984> − − <ref name=Hald1984> − − < ref name = hald1984 > − − {{Citation − − {{Citation − − {引文 − − |last1=Hald |first1=Anders − − |last1=Hald |first1=Anders − − 1 = Hald | first1 = Anders − − |last2=de Moivre |first2=Abraham − − |last2=de Moivre |first2=Abraham − − 2 = de Moivre | first2 = Abraham − − |last3=McClintock |first3=Bruce − − |last3=McClintock |first3=Bruce − − 3 = McClintock | first3 = Bruce − − |title=A. de Moivre: 'De Mensura Sortis' or 'On the Measurement of Chance' − − |title=A. de Moivre: 'De Mensura Sortis' or 'On the Measurement of Chance' − − | title = a.‘ De Mensura Sortis’或‘ On the Measurement of Chance’ − − |journal=International Statistical Review / Revue Internationale de Statistique − − |journal=International Statistical Review / Revue Internationale de Statistique − − 国际统计评论/国际统计杂志 − − |year=1984 |volume=52 |issue=3 |pages=229–262 − − |year=1984 |volume=52 |issue=3 |pages=229–262 − − 1984 | volume = 52 | issue = 3 | pages = 229-262 − − |doi=10.2307/1403045 − − |doi=10.2307/1403045 − − 10.2307/1403045 − − |jstor=1403045 − − |jstor=1403045 − − 1403045 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Newcomb1860> − − <ref name=Newcomb1860> − − < ref name = newcomb1860 > − − {{citation − − {{citation − − { citation − − |last1=Newcomb − − |last1=Newcomb − − 1 = Newcomb − − |first1=Simon − − |first1=Simon − − 1 = Simon − − |year=1860 − − |year=1860 − − 1860年 − − |title=Notes on the theory of probabilities − − |title=Notes on the theory of probabilities − − | title = 概率论笔记 − − |journal=The Mathematical Monthly − − |journal=The Mathematical Monthly − − | journal = The Mathematical Monthly − − |volume=2 − − |volume=2 − − 2 − − |issue=4 − − |issue=4 − − 第四期 − − |pages=134–140 − − |pages=134–140 − − | 页数 = 134-140 − − |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433069075590&seq=150 − − |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433069075590&seq=150 − − Https://babel.hathitrust.org/cgi/pt?id=nyp.33433069075590&seq=150 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=vonBortkiewitsch1898> − − <ref name=vonBortkiewitsch1898> − − < ref name = vonbortkiewitsch1898 > − − {{citation − − {{citation − − { citation − − |last1=von Bortkiewitsch |first1=Ladislaus <!-- This is the spelling of the name as it appears in the book, the Polish version would be Vladislav Bortkevič--> − − |last1=von Bortkiewitsch |first1=Ladislaus <!-- This is the spelling of the name as it appears in the book, the Polish version would be Vladislav Bortkevič--> − − 1 = von Bortkiewitsch | first1 = Ladislaus < ! ——这是这个名字在书中出现时的拼写，波兰语版本是 vladislav bortkevi —— > − − |title=Das Gesetz der kleinen Zahlen − − |title=Das Gesetz der kleinen Zahlen − − |title=Das Gesetz der kleinen Zahlen − − |trans-title=The law of small numbers − − |trans-title=The law of small numbers − − | trans-title = 小数定律 − − |language=German − − |language=German − − 语言 = 德语 − − |publisher=B. G. Teubner |location=Leipzig, Germany − − |publisher=B. G. Teubner |location=Leipzig, Germany − − 德国莱比锡 − − |year=1898 − − |year=1898 − − 1898年 − − |page=On [https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/13 page 1], Bortkiewicz presents the Poisson distribution. On [https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/35 pages 23–25], Bortkiewitsch presents his analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preußischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.) − − |page=On [https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/13 page 1], Bortkiewicz presents the Poisson distribution. On [https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/35 pages 23–25], Bortkiewitsch presents his analysis of "4. Beispiel: Die durch Schlag eines Pferdes im preußischen Heere Getöteten." (4. Example: Those killed in the Prussian army by a horse's kick.) − − 在纽约 https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/13第一页，Bortkiewicz 展示了纽约泊松分佈。在《 https://digibus.ub.uni-stuttgart.de/viewer/object/1543508614348/3523-25页，Bortkiewitsch 提出了他对《4》的分析。Beispiel: Die durch Schlag eines Pferdes im preußischen Heere Getöteten."(4.那些在普鲁士军队中被马踢死的人 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Lehmann1986> − − <ref name=Lehmann1986> − − < ref name = lehmann1986 > − − {{citation − − {{citation − − { citation − − |last1=Lehmann |first1=Erich Leo − − |last1=Lehmann |first1=Erich Leo − − 1 = Lehmann | first1 = Erich Leo − − |title=Testing Statistical Hypotheses − − |title=Testing Statistical Hypotheses − − | title = 检验统计假说 − − |publisher=Springer Verlag |location=New York, NJ, USA − − |publisher=Springer Verlag |location=New York, NJ, USA − − 纽约，新泽西，美国 − − |edition=second − − |edition=second − − 第二季，第二集 − − |year=1986 − − |year=1986 − − 1986年 − − |isbn=978-0-387-94919-2}}</ref> − − |isbn=978-0-387-94919-2}}</ref> − − 978-0-387-94919-2}} </ref > − − <ref name=Yates2014> − − <ref name=Yates2014> − − < ref name = yates2014 > − − {{citation − − {{citation − − { citation − − |last1=Yates |first1=Roy D. |last2=Goodman |first2=David J. − − |last1=Yates |first1=Roy D. |last2=Goodman |first2=David J. − − 1 = Yates | first1 = Roy d | last2 = Goodman | first2 = David j. − − |title=Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers − − |title=Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers − − | title = 概率和随机过程: 电气和计算机工程师的友好介绍 − − |year=2014 − − |year=2014 − − 2014年 − − |publisher=Wiley − − |publisher=Wiley − − | publisher = Wiley − − |location=Hoboken, USA − − |location=Hoboken, USA − − | 地点: 霍博肯 − − |edition=2nd |isbn=978-0-471-45259-1}} − − |edition=2nd |isbn=978-0-471-45259-1}} − − | edition = 2 nd | isbn = 978-0-471-45259-1} − − </ref> − − </ref> − − </ref > − − <ref name=Choi1994> − − <ref name=Choi1994> − − < ref name = choi1994 > − − {{citation − − {{citation − − { citation − − |last1=Choi |first1=Kwok P. − − |last1=Choi |first1=Kwok P. − − 1 = Choi | first1 = Kwok p. − − |title=On the medians of gamma distributions and an equation of Ramanujan − − |title=On the medians of gamma distributions and an equation of Ramanujan − − | title = 关于 γ 分布的中线和 Ramanujan 的一个方程 − − |journal=Proceedings of the American Mathematical Society − − |journal=Proceedings of the American Mathematical Society − − | journal = Proceedings of the American Mathematical Society − − |year=1994 |volume=121 |issue=1 |pages=245–251 − − |year=1994 |volume=121 |issue=1 |pages=245–251 − − 1994 | volume = 121 | issue = 1 | pages = 245-251 − − |doi=10.2307/2160389 − − |doi=10.2307/2160389 − − 10.2307/2160389 − − |jstor=2160389|doi-access=free}}</ref> − − |jstor=2160389|doi-access=free}}</ref> − − 2160389 | doi-access = free } </ref > − − <ref 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D. − − 1 = Clarke | first1 = r.D. − − |title=An application of the Poisson distribution − − |title=An application of the Poisson distribution − − | title = 泊松分佈的一个应用程序 − − |journal=Journal of the Institute of Actuaries − − |journal=Journal of the Institute of Actuaries − − | Journal = the Journal of the Institute of Actuaries − − |year=1946 |volume=72 |issue=3 |page=481 − − |year=1946 |volume=72 |issue=3 |page=481 − − 1946 | volume = 72 | issue = 3 | page = 481 − − |doi=10.1017/S0020268100035435 − − |doi=10.1017/S0020268100035435 − − | doi = 10.1017/S0020268100035435 − − |url=https://www.actuaries.org.uk/system/files/documents/pdf/0481.pdf − − |url=https://www.actuaries.org.uk/system/files/documents/pdf/0481.pdf − − Https://www.actuaries.org.uk/system/files/documents/pdf/0481.pdf − − |doi-access=free − − |doi-access=free − − 免费访问 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Gallagher1976> − − <ref name=Gallagher1976> − − < ref name = gallagher 1976 > − − {{Citation − − {{Citation − − {引文 − − |last=Gallagher |first=Patrick X. − − |last=Gallagher |first=Patrick X. − − 最后 = 加拉格尔 | 第一 = 帕特里克 x。 − − |title=On the distribution of primes in short intervals − − |title=On the distribution of primes in short intervals − − | title = 关于素数在短时间内的分布 − − |journal=Mathematika − − |journal=Mathematika − − 日志 = mathematica − − |year=1976 |volume=23 |issue=1 |pages=4–9 − − |year=1976 |volume=23 |issue=1 |pages=4–9 − − 1976 | volume = 23 | issue = 1 | pages = 4-9 − − |doi=10.1112/s0025579300016442 − − |doi=10.1112/s0025579300016442 − − | doi = 10.1112/s0025579300016442 − − |doi-access=free − − |doi-access=free − − 免费访问 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Hardy1923> − − <ref name=Hardy1923> − − < ref name = hardy1923 > − − {{Citation − − {{Citation − − {引文 − − |last1=Hardy |first1=Godfrey H. |author-link1=G. 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first = Francis y. − − |authorlink=Francis Ysidro Edgeworth − − |authorlink=Francis Ysidro Edgeworth − − 作者/链接 = 弗朗西斯·伊西德罗·埃奇沃思 − − |title=On the use of the theory of probabilities in statistics relating to society − − |title=On the use of the theory of probabilities in statistics relating to society − − 论概率论在与社会相关的统计学中的应用 − − |journal=[[Journal of the Royal Statistical Society]] − − |journal=Journal of the Royal Statistical Society − − 英国皇家统计学会杂志 − − |year=1913 |volume=76 |issue=2 |pages=165–193 − − |year=1913 |volume=76 |issue=2 |pages=165–193 − − 1913 | volume = 76 | issue = 2 | pages = 165-193 − − |jstor=2340091 |doi=10.2307/2340091 − − |jstor=2340091 |doi=10.2307/2340091 − − | jstor = 2340091 | doi = 2340091 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Knuth1997> − − <ref name=Knuth1997> − − < ref name = knuth1997 > − − {{citation − − {{citation − − { citation − − |last=Knuth |first=Donald Ervin − − |last=Knuth |first=Donald Ervin − − 最后 = Knuth | first = Donald Ervin − − |title = Seminumerical Algorithms − − |title = Seminumerical Algorithms − − 半数算法 − − |publisher=[[Addison Wesley]] − − |publisher=Addison Wesley − − 艾迪生 · 韦斯利 − − |series=[[The Art of Computer Programming]] − − |series=The Art of Computer Programming − − 计算机编程的艺术 − − |volume=2 − − |volume=2 − − 2 − − |edition=3rd − − |edition=3rd − − 3 rd − − |year=1997 − − |year=1997 − − 1997年 − − |isbn=978-0-201-89684-8 − − |isbn=978-0-201-89684-8 − − | isbn = 978-0-201-89684-8 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name="Devroye1986"> − − <ref name="Devroye1986"> − − < ref name ="devroye1986"> − − {{citation − − {{citation − − { citation − − |last=Devroye |first=Luc |author-link=Luc Devroye − − |last=Devroye |first=Luc |author-link=Luc Devroye − − 最后 = Devroye | first = Luc | author-link = Luc Devroye − − |chapter=Discrete Univariate Distributions − − |chapter=Discrete Univariate Distributions − − | chapter = 离散单变量分布 − − |chapterurl=http://luc.devroye.org/chapter_ten.pdf − − 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|first2=Nick E. − − |first2=Nick E. − − 2 = Nick e. − − |authorlink2=Nick Day − − |authorlink2=Nick Day − − 2 = Nick Day − − |title=Statistical Methods in Cancer Research: Volume 2—The Design and Analysis of Cohort Studies − − |title=Statistical Methods in Cancer Research: Volume 2—The Design and Analysis of Cohort Studies − − 癌症研究的统计方法: 第2卷ー队列研究的设计与分析 − − |year=1987 − − |year=1987 − − 1987年 − − |publisher=[[International Agency for Research on Cancer]] − − |publisher=International Agency for Research on Cancer − − 国际癌症研究机构 − − |location=Lyon, France − − |location=Lyon, France − − | 地点: 法国里昂 − − |url=http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php − − |url=http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php − − Http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php − − |isbn=978-92-832-0182-3 − − |isbn=978-92-832-0182-3 − − | isbn = 978-92-832-0182-3 − − |access-date=2012-03-11 − − |access-date=2012-03-11 − − | access-date = 2012-03-11 − − |archive-url=https://web.archive.org/web/20180808161401/http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php − − |archive-url=https://web.archive.org/web/20180808161401/http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php − − | archive-url = https://web.archive.org/web/20180808161401/http://www.iarc.fr/en/publications/pdfs-online/stat/sp82/index.php − − |archive-date=2018-08-08 − − |archive-date=2018-08-08 − − | archive-date = 2018-08-08 − − |url-status=dead − − |url-status=dead − − 地位 = 死亡 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Fink1976> − − <ref name=Fink1976> − − < ref name = fink1976 > − − {{citation − − {{citation − − { citation − − |last=Fink |first=Daniel − − |last=Fink |first=Daniel − − | last = Fink | first = Daniel − − |title=A Compendium of Conjugate Priors − − |title=A Compendium of Conjugate Priors − − | 题目 = 共轭先验概要 − − |year=1997 − − |year=1997 − − 1997年 − − |publisher= − − |publisher= − − | publisher = − − |location= − − |location= − − | location = − − |isbn= − − |isbn= − − | isbn = − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Gelman2003> − − <ref name=Gelman2003> − − < ref name = gelman2003 > − − {{citation − − {{citation − − { citation − − |last1=Gelman |last2=Carlin |first2=John B. − − |last1=Gelman |last2=Carlin |first2=John B. − − 1 = Gelman | last2 = Carlin | first2 = John b. − − |last3=Stern |first3=Hal S. − − |last3=Stern |first3=Hal S. − − 3 = Stern | first3 = Hal s. − − |last4=Rubin |first4=Donald B. − − |last4=Rubin |first4=Donald B. − − 4 = Rubin | first4 = Donald b. − − |title=Bayesian Data Analysis − − |title=Bayesian Data Analysis − − | title = Bayesian Data Analysis − − |year=2003 − − |year=2003 − − 2003年 − − |edition=2nd − − |edition=2nd − − 2nd − − |publisher=Chapman & Hall/CRC − − |publisher=Chapman & Hall/CRC − − | publisher = Chapman & Hall/CRC − − |location=Boca Raton, FL, USA − − |location=Boca Raton, FL, USA − − | location = Boca Raton，FL，USA − − |isbn=1-58488-388-X − − |isbn=1-58488-388-X − − | isbn = 1-58488-388-X − − |doi= − − |doi= − − | doi = − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Clevenson1975> − − <ref name=Clevenson1975> − − < ref name = clevenson1975 > − − {{citation − − {{citation − − { citation − − |last1=Clevenson |first1=M. 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Lawrence − − 1 = Clevenson | first1 = m.劳伦斯 − − |last2=Zidek |first2=James V. − − |last2=Zidek |first2=James V. − − 2 = zidk | first2 = James v. − − |title=Simultaneous Estimation of the Means of Independent Poisson Laws − − |title=Simultaneous Estimation of the Means of Independent Poisson Laws − − | title = 独立泊松定律均值的同时估计 − − |journal=Journal of the American Statistical Association − − |journal=Journal of the American Statistical Association − − 美国统计协会杂志 − − |year=1975 |volume=70 |issue=351 |pages=698–705 − − |year=1975 |volume=70 |issue=351 |pages=698–705 − − 1975 | volume = 70 | issue = 351 | pages = 698-705 − − |doi=10.1080/01621459.1975.10482497 − − |doi=10.1080/01621459.1975.10482497 − − | doi = 10.1080/01621459.1975.10482497 − − |jstor=2285958 − − |jstor=2285958 − − 2285958 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Berger1985> − − <ref name=Berger1985> − − < ref name = berger1985 > − − {{citation − − {{citation − − { citation − − |last=Berger |first=James O. − − |last=Berger |first=James O. − − 最后 = 伯格 | 第一 = 詹姆斯 o。 − − |title=Statistical Decision Theory and Bayesian Analysis − − |title=Statistical Decision Theory and Bayesian Analysis − − 统计决策理论和贝叶斯分析 − − |series=Springer Series in Statistics − − |series=Springer Series in Statistics − − | 系列 = 统计学中的斯普林格系列 − − |year=1985 − − |year=1985 − − 1985年 − − |edition=2nd − − |edition=2nd − − 2nd − − |publisher=Springer-Verlag − − |publisher=Springer-Verlag − − | publisher = Springer-Verlag − − |location=New York, NJ, USA − − |location=New York, NJ, USA − − 纽约，新泽西，美国 − − |isbn=978-0-387-96098-2 − − |isbn=978-0-387-96098-2 − − | isbn = 978-0-387-96098-2 − − |doi=10.1007/978-1-4757-4286-2 − − |doi=10.1007/978-1-4757-4286-2 − − | doi = 10.1007/978-1-4757-4286-2 − − }}</ref> − − }}</ref> − − } </ref > − − <ref name=Loukas1986> − − <ref name=Loukas1986> − − < ref name = loukas1986 > − − {{citation − − {{citation − − { citation − − |last1=Loukas |first=Sotirios − − |last1=Loukas |first=Sotirios − − 1 = Loukas | first = Sotirios − − |last2=Kemp |first2=C. David − − |last2=Kemp |first2=C. David − − 2 = Kemp | first2 = c.大卫 − − |title=The Index of Dispersion Test for the Bivariate Poisson Distribution − − |title=The Index of Dispersion Test for the Bivariate Poisson Distribution − − | title = 二元泊松分佈的离散度检验指数 − − |journal=Biometrics − − |journal=Biometrics − − 生物测定学 − − |year=1986 |volume=42 |issue=4 |pages=941–948 − − |year=1986 |volume=42 |issue=4 |pages=941–948 − − | year = 1986 | volume = 42 | issue = 4 | pages = 941-948 − − |doi= 10.2307/2530708 − − |doi= 10.2307/2530708 − − 10.2307/2530708 − − |jstor=2530708 − − |jstor=2530708 − − 2530708 − − }}</ref> − − }}</ref> − − } </ref > − − }} − − }} − − }} − − − − === Sources资源 === − − {{refbegin}} − − * {{citation − − |last1 = Ahrens |first1 = Joachim H. − − |last1 = Ahrens |first1 = Joachim H. − − 1 = Ahrens | first1 = Joachim h. − − |last2 = Dieter |first2 = Ulrich − − |last2 = Dieter |first2 = Ulrich − − 2 = Dieter | first2 = Ulrich − − |title = Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions − − |title = Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions − − | title = 从伽马分布、贝塔分布、泊松分布和二项分布抽样的计算机方法 − − |journal = Computing − − |journal = Computing − − 日志 = 计算 − − |year = 1974 − − |year = 1974 − − 1974年 − − |volume = 12 − − |volume = 12 − − 12 − − |issue = 3 − − |issue = 3 − − 第三期 − − |pages = 223–246 − − |pages = 223–246 − − | page = 223-246 − − |doi = 10.1007/BF02293108 − − |doi = 10.1007/BF02293108 − − | doi = 10.1007/BF02293108 − − }} − − }} − − }} − − * {{citation − − |last1 = Ahrens |first1 = Joachim H. − − |last1 = Ahrens |first1 = Joachim H. − − 1 = Ahrens | first1 = Joachim h. − − |last2 = Dieter |first2 = Ulrich − − |last2 = Dieter |first2 = Ulrich − − 2 = Dieter | first2 = Ulrich − − |title = Computer Generation of Poisson Deviates − − |title = Computer Generation of Poisson Deviates − − | title = 计算机生成的泊松偏离 − − |journal = ACM Transactions on Mathematical Software − − |journal = ACM Transactions on Mathematical Software − − | journal = ACM 数学软件汇刊 − − |year = 1982 − − |year = 1982 − − 1982年 − − |volume = 8 − − |volume = 8 − − 8 − − |issue = 2 − − |issue = 2 − − 2 − − |pages = 163–179 − − |pages = 163–179 − − | 页 = 163-179 − − − − − − 10.1145/355993.355997 − − }}+ − + − + − + − + − + − |last1 = Evans |first1 = Ronald J.+ − − − − 1 = Evans | first1 = Ronald j. − − |last2 = Boersma |first2 = J. − − − − 2 = Boersma | first2 = j. − − |last3 = Blachman |first3 = N. M. − − |last3 = Blachman |first3 = N. M. − − 3 = Blachman | first3 = n. − − |last4 = Jagers |first4 = A. A. − − |last4 = Jagers |first4 = A. A. − − 4 = Jagers | first4 = a.答:。 − − |title = The Entropy of a Poisson Distribution: Problem 87-6 − − |title = The Entropy of a Poisson Distribution: Problem 87-6 − − | title = 泊松分佈的熵: 问题87-6 − − |journal = SIAM Review − − |journal = SIAM Review − − | journal = SIAM Review − − |year = 1988 − − |year = 1988 − − 1988年 − − |volume = 30 − − |volume = 30 − − 30 − − |issue = 2 − − |issue = 2 − − 2 − − |pages = 314–317 − − |pages = 314–317 − − | 页数 = 314-317 − − |doi = 10.1137/1030059 − − |doi = 10.1137/1030059 − − 10.1137/1030059 − − |url = https://research.tue.nl/nl/publications/solution-to-problem-876--the-entropy-of-a-poisson-distribution(94cf6dd2-b35e-41c8-9da7-6ec69ca391a0).html − − − − Https://research.tue.nl/nl/publications/solution-to-problem-876--the-entropy-of-a-poisson-distribution(94cf6dd2-b35e-41c8-9da7-6ec69ca391a0).html − − }} − − }} − − }} − − {{-}}+ − + − {{ProbDistributions|discrete-infinite}}+ − + − − − {{Authority control}} − − − − − − − − Category:Articles with example pseudocode − − 类别: 带有伪代码示例的文章 − − − − Category:Conjugate prior distributions − − 范畴: 共轭先验分布 − − − − Category:Factorial and binomial topics − − 类别: 阶乘和二项式主题 − − [[Category:Infinitely divisible probability distributions]] − − Category:Infinitely divisible probability distributions − − 类别: 无限可分的概率分布 − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Poisson distribution]]. Its edit history can be viewed at [[泊松分布/edithistory]]</small></noinclude> − − [[Category:待整理页面]]
无编辑摘要
审校：fairywang
审校：fairywang
In [[probability theory]] and [[statistics]], the '''Poisson distribution''' ({{IPAc-en|'|p|w|ɑː|s|ɒ|n}}; {{IPA-fr|pwasɔ̃}}), named after [[France|French]] mathematician [[Siméon Denis Poisson]], is a [[discrete probability distribution]] that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and [[Statistical independence|independently]] of the time since the last event.{{r|Haight1967}} The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
In [[probability theory]] and [[statistics]], the '''Poisson distribution''' ({{IPAc-en|'|p|w|ɑː|s|ɒ|n}}; {{IPA-fr|pwasɔ̃}}), named after [[France|French]] mathematician [[Siméon Denis Poisson]], is a [[discrete probability distribution]] that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and [[Statistical independence|independently]] of the time since the last event.{{r|Haight1967}} The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
In probability theory and statistics, the Poisson distribution (; ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
In probability theory and statistics, the Poisson distribution (; ), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
在概率论和统计学中，'''<font color="#ff8000"> 泊松分布 Poisson distribution</font>'''是以法国数学家西莫恩·德尼·泊松 Siméon Denis Poisson命名的，是一个离散的概率分布，它表示在一个固定的时间段或空间中一定数量的事件的发生概率，这些事件以一个已知的常数平均速率发生，并且独立于与上一个事件的间隔发生时间。还可以用来表示其他有特定间隔的事件数量，如距离、面积或体积。
在概率论和统计学中，''' 泊松分布 Poisson distribution'''是以法国数学家西莫恩·德尼·泊松 Siméon Denis Poisson命名的，是一个离散的概率分布，它表示在一个固定的时间段或空间中一定数量的事件的发生概率，这些事件以一个已知的常数平均速率发生，并且独立于与上一个事件的间隔发生时间。还可以用来表示其他有特定间隔的事件数量，如距离、面积或体积。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“它表示在一个固定的时间段或空间中一定数量的事件的发生概率”改为“它表示在一个固定的时间段或空间中，一定数量的事件发生的概率”
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“它表示在一个固定的时间段或空间中一定数量的事件的发生概率”改为“它表示在一个固定的时间段或空间中，一定数量的事件发生的概率”
For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received in a day obeys a Poisson distribution. Other examples that may follow a Poisson distribution include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.
For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. If receiving any particular piece of mail does not affect the arrival times of future pieces of mail, i.e., if pieces of mail from a wide range of sources arrive independently of one another, then a reasonable assumption is that the number of pieces of mail received in a day obeys a Poisson distribution. Other examples that may follow a Poisson distribution include the number of phone calls received by a call center per hour and the number of decay events per second from a radioactive source.
例如，记录每天收到邮件数量的个人可能会注意到，他们平均每天收到4封信。如果收到任何邮件都并不影响未来邮件的到达时间，也就是说，如果不同来源的邮件彼此独立地到达，那么一个合理的假设是，每天收到的邮件数量服从一个'''<font color="#ff8000">泊松分布</font>'''。其他可能遵循一个'''<font color="#ff8000">泊松分布</font>'''的例子包括：呼叫中心每小时接到的电话数量和每秒从放射源衰变事件的数量。
例如，记录每天收到邮件数量的个人可能会注意到，他们平均每天收到4封信。如果收到任何邮件都并不影响未来邮件的到达时间，也就是说，如果不同来源的邮件彼此独立地到达，那么一个合理的假设是，每天收到的邮件数量服从一个'''泊松分布'''。其他可能遵循一个'''泊松分布'''的例子包括：呼叫中心每小时接到的电话数量和每秒从放射源衰变事件的数量。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“呼叫中心每小时接到的电话数量和每秒从放射源衰变事件的数量。”改为“呼叫中心每小时接到的电话数量和每秒从放射源的衰变数。”
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“呼叫中心每小时接到的电话数量和每秒从放射源衰变事件的数量。”改为“呼叫中心每小时接到的电话数量和每秒从放射源的衰变数。”
The Poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space.
The Poisson distribution is popular for modeling the number of times an event occurs in an interval of time or space.
'''<font color="#ff8000">泊松分布</font>'''模型用来模拟一个事件在一段时间或空间内发生的次数。
'''泊松分布'''模型用来模拟一个事件在一段时间或空间内发生的次数。
If these conditions are true, then is a Poisson random variable, and the distribution of is a Poisson distribution.
If these conditions are true, then is a Poisson random variable, and the distribution of is a Poisson distribution.
如果这些条件成立，那么它是一个泊松随机变量，其分布是一个'''<font color="#ff8000"> {泊松分布 Poisson distribution</font>'''。
如果这些条件成立，那么它是一个泊松随机变量，其分布是一个''' {泊松分布 Poisson distribution'''。
The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals divided by the number of trials, as the number of trials approaches infinity (see Related distributions).
The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals divided by the number of trials, as the number of trials approaches infinity (see Related distributions).
每次试验的成功概率除以总试验次数，随着试验的数量趋于无穷大，泊松分布也是'''<font color="#ff8000"> 二项式分布Binomial distribution</font>'''的极限。（可参考相关分布）
每次试验的成功概率除以总试验次数，随着试验的数量趋于无穷大，泊松分布也是''' 二项式分布Binomial distribution'''的极限。（可参考相关分布）
===Probability of events for a Poisson distribution'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''的事件概率===
===Probability of events for a Poisson distribution''' {泊松分布 Poisson distribution'''的事件概率===
An event can occur 0, 1, 2, ... times in an interval. The average number of events in an interval is designated <math> \lambda </math> (lambda). <math> \lambda </math> is the event rate, also called the rate parameter. The probability of observing {{mvar|k}} events in an interval is given by the equation
An event can occur 0, 1, 2, ... times in an interval. The average number of events in an interval is designated <math> \lambda </math> (lambda). <math> \lambda </math> is the event rate, also called the rate parameter. The probability of observing {{mvar|k}} events in an interval is given by the equation
An event can occur 0, 1, 2, ... times in an interval. The average number of events in an interval is designated <math> \lambda </math> (lambda). <math> \lambda </math> is the event rate, also called the rate parameter. The probability of observing events in an interval is given by the equation
An event can occur 0, 1, 2, ... times in an interval. The average number of events in an interval is designated <math> \lambda </math> (lambda). <math> \lambda </math> is the event rate, also called the rate parameter. The probability of observing events in an interval is given by the equation
一个事件可以在一个间隔内发生0,1,2，... 次。区间内的平均事件数被指定为 < math > lambda </math > (lambda)。Lambda </math > 是'''<font color="#ff8000">事件速率Event rate </font>'''，也称为'''<font color="#ff8000"> 速率参数Rate parameter</font>'''。以下方程给出了在一个区间内观测事件的概率
一个事件可以在一个间隔内发生0,1,2，... 次。区间内的平均事件数被指定为 < math > lambda </math > (lambda)。Lambda </math > 是'''事件速率Event rate '''，也称为''' 速率参数Rate parameter'''。以下方程给出了在一个区间内观测事件的概率
==== Examples of probability for Poisson distributions'''<font color="#ff8000"> {泊松分佈 Poisson distribution</font>'''概率的示例 ====
==== Examples of probability for Poisson distributions''' {泊松分布 Poisson distribution'''概率的示例 ====
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).
每分钟抵达学生会的学生人数可能不会遵循一个'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''，因为这个比率不是恒定的(上课时间的低比率，课间时的高比率) ，而且每个学生的到达也不是独立的(学生往往是成群结队来的)。
每分钟抵达学生会的学生人数可能不会遵循一个''' 泊松分布Poisson distribution.'''，因为这个比率不是恒定的(上课时间的低比率，课间时的高比率) ，而且每个学生的到达也不是独立的(学生往往是成群结队来的)。
The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.
The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.
一次大的强震会增加发生类似震级余震的可能性，那么一个国家每年发生5级地震的次数可能不会服从'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''。
一次大的强震会增加发生类似震级余震的可能性，那么一个国家每年发生5级地震的次数可能不会服从''' 泊松分布Poisson distribution.'''。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a [[Zero-truncated Poisson distribution]].
Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a [[Zero-truncated Poisson distribution]].
Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a Zero-truncated Poisson distribution.
Examples in which at least one event is guaranteed are not Poission distributed; but may be modeled using a Zero-truncated Poisson distribution.
至少有一个事件确定发生的情况不是 Poission 分布式的，但也许可以使用零截断'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''进行建模。
至少有一个事件确定发生的情况不是 Poission 分布式的，但也许可以使用零截断''' 泊松分布Poisson distribution.'''进行建模。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a [[Zero-inflated model]].
Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a [[Zero-inflated model]].
=== '''<font color="#ff8000"> 描述统计学Descriptive statistics</font>'''===
=== ''' 描述统计学Descriptive statistics'''===
* The [[expected value]] and [[variance]] of a Poisson-distributed random variable are both equal to λ.
* The [[expected value]] and [[variance]] of a Poisson-distributed random variable are both equal to λ.
*一个具有非整值λ 的泊松分布随机变量的统计值等于<math>\scriptstyle\lfloor \lambda \rfloor</math>, 小于''λ''的最大整数。它也写作[[floor function|floor]](λ). λ为正整数时，取值为''λ'' 以及 ''λ'' − 1。
*一个具有非整值λ 的泊松分布随机变量的统计值等于<math>\scriptstyle\lfloor \lambda \rfloor</math>, 小于''λ''的最大整数。它也写作[[floor function|floor]](λ). λ为正整数时，取值为''λ'' 以及 ''λ'' − 1。
*所有泊松分布的积均等于期望值 ''λ''。泊松分布的n阶指数积为''λ''<sup>''n''</sup>。
*所有泊松分布的积均等于期望值 ''λ''。泊松分布的n阶指数积为''λ''<sup>''n''</sup>。
*期望值与泊松过程有时分解为“强度”与“面积”的乘积（或更一般地表示为强度函数随时间或空间的积分，有时描述为'''<font color="#32CD32">“暴露”“exposure”</font>'''。）{{r|Helske2017}}
*期望值与泊松过程有时分解为“强度”与“面积”的乘积（或更一般地表示为强度函数随时间或空间的积分，有时描述为'''<font color="#32CD32">“暴露”“exposure”'''。）{{r|Helske2017}}
=== Median 中值===
=== Median 中值===
where the {braces} denote Stirling numbers of the second kind. The coefficients of the polynomials have a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, then Dobinski's formula says that the nth moment equals the number of partitions of a set of size n.
where the {braces} denote Stirling numbers of the second kind. The coefficients of the polynomials have a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, then Dobinski's formula says that the nth moment equals the number of partitions of a set of size n.
其中{括号}表示第二类 Stirling 数。多项式的系数具有组合意义。事实上，当泊松分佈的期望值是1时，那么 Dobinski 的公式说第 n 个时刻等于一组大小为 n 的分区的数目。
其中{括号}表示第二类 Stirling 数。多项式的系数具有组合意义。事实上，当泊松分布的期望值是1时，那么 Dobinski 的公式说第 n 个时刻等于一组大小为 n 的分区的数目。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
For the non-centered moments we define <math>B=k/\lambda</math>, then{{r|Jagadeesan2017}}
For the non-centered moments we define <math>B=k/\lambda</math>, then{{r|Jagadeesan2017}}
If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are independent, then <math>\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right)</math>. A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.
If <math>X_i \sim \operatorname{Pois}(\lambda_i)</math> for <math>i=1,\dotsc,n</math> are independent, then <math>\sum_{i=1}^n X_i \sim \operatorname{Pois}\left(\sum_{i=1}^n \lambda_i\right)</math>. A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.
如果对于 < math > i = 1，dotsc，n </math > 是独立的，那么 < math > sum { i = 1} ^ n xi sim 操作者名{ Pois }左(sum { i = 1} ^ n lambda _ i 右) </math > 。一个逆定理是雷科夫定理，它说如果两个独立的随机变量之和是'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''的，那么这两个独立的随机变量之和也是'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''的。
如果对于 < math > i = 1，dotsc，n </math > 是独立的，那么 < math > sum { i = 1} ^ n xi sim 操作者名{ Pois }左(sum { i = 1} ^ n lambda _ i 右) </math > 。一个逆定理是雷科夫定理，它说如果两个独立的随机变量之和是''' 泊松分布Poisson distribution.'''的，那么这两个独立的随机变量之和也是''' 泊松分布Poisson distribution.'''的。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
=== Other properties 其他特性===
=== Other properties 其他特性===
*泊松分布是[[无限可除性（概率）|无限可除]] 概率分布，{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}}。
*泊松分布是[[无限可除性（概率）|无限可除]] 概率分布，{{r|Laha1979|p=233}}{{r|Johnson2005|p=164}}。
*<math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> 的直接'''<font color="#ff8000"> 相对熵（K-L散度）[[Kullback–Leibler divergence]]</font>'''由以下给出：
*<math>\operatorname{Pois}(\lambda_0)</math> from <math>\operatorname{Pois}(\lambda)</math> 的直接''' 相对熵（K-L散度）[[Kullback–Leibler divergence]]'''由以下给出：
:: <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>
:: <math>\operatorname{D}_{\text{KL}}(\lambda\mid\lambda_0) = \lambda_0 - \lambda + \lambda \log \frac{\lambda}{\lambda_0}.</math>
* Bounds for the tail probabilities of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> can be derived using a [[Chernoff bound]] argument.{{r|Mitzenmacher2005|p=97-98}}
* Bounds for the tail probabilities of a Poisson random variable <math> X \sim \operatorname{Pois}(\lambda)</math> can be derived using a [[Chernoff bound]] argument.{{r|Mitzenmacher2005|p=97-98}}
*泊松随机变量尾概率的界<math> X \sim \operatorname{Pois}(\lambda)</math> 可以用[['''<font color="#ff8000"> 切诺夫界Chernoff bound</font>''']]参数派生{{r|Mitzenmacher2005|p=97-98}}
*泊松随机变量尾概率的界<math> X \sim \operatorname{Pois}(\lambda)</math> 可以用[[''' 切诺夫界Chernoff bound''']]参数派生{{r|Mitzenmacher2005|p=97-98}}
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,
:: <math> P(X \geq x) \leq \frac{(e \lambda)^x e^{-\lambda}}{x^x}, \text{ for } x > \lambda</math>,
The upper bound is proved using a standard Chernoff bound.
The upper bound is proved using a standard Chernoff bound.
利用标准的'''<font color="#ff8000"> 切诺夫界Chernoff bound</font>'''证明了上界的存在性。
利用标准的''' 切诺夫界Chernoff bound'''证明了上界的存在性。
== '''<font color="#ff8000"> 相关分布'Related distributions</font>''==
== ''' 相关分布'Related distributions''==
===Genera通常l===
===Genera通常l===
==='''<font color="#ff8000"> Poisson Approximation 泊松近似</font>'''===
===''' Poisson Approximation 泊松近似'''===
=== '''<font color="#ff8000"> 二元泊松分布Bivariate Poisson distribution</font>'''===
=== ''' 二元泊松分布Bivariate Poisson distribution'''===
This distribution has been extended to the [[bivariate]] case.{{r|Loukas1986}} The [[generating function]] for this distribution is
This distribution has been extended to the [[bivariate]] case.{{r|Loukas1986}} The [[generating function]] for this distribution is
A simple way to generate a bivariate Poisson distribution <math>X_1,X_2</math> is to take three independent Poisson distributions <math>Y_1,Y_2,Y_3</math> with means <math>\lambda_1,\lambda_2,\lambda_3</math> and then set <math>X_1 = Y_1 + Y_3,X_2 = Y_2 + Y_3</math>. The probability function of the bivariate Poisson distribution is
A simple way to generate a bivariate Poisson distribution <math>X_1,X_2</math> is to take three independent Poisson distributions <math>Y_1,Y_2,Y_3</math> with means <math>\lambda_1,\lambda_2,\lambda_3</math> and then set <math>X_1 = Y_1 + Y_3,X_2 = Y_2 + Y_3</math>. The probability function of the bivariate Poisson distribution is
一个简单的方法来产生一个二变量的泊松分布分布: 取3个独立的 Poisson 分布 < math > y _ 1，y _ 2，y _ 3 </math > ，用 < math > lambda _ 1，lambda _ 2，lambda _ 3 </math > 然后设置 < math > x _ 1 = y _ 1 + y _ 3，x _ 2 = y _ 2 + y _ 3 </math > 。二元概率密度函数变量的泊松分布变量是
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈分布”改为“泊松分布”，“泊松分佈变量”改为“泊松分布变量”
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布分布”改为“泊松分布”，“泊松分布变量”改为“泊松分布变量”
: <math>
: <math>
==='''<font color="#ff8000"> 自由泊松分布Free Poisson distribution</font>'''===
===''' 自由泊松分布Free Poisson distribution'''===
The free Poisson distribution<ref>Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992</ref> with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in [[free probability]] theory as the limit of repeated [[free convolution]]
The free Poisson distribution<ref>Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992</ref> with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in [[free probability]] theory as the limit of repeated [[free convolution]]
The free Poisson distribution with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in free probability theory as the limit of repeated free convolution
The free Poisson distribution with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in free probability theory as the limit of repeated free convolution
带有跳跃大小 < math > alpha </math > 和速率 < math > lambda </math > 的'''<font color="#ff8000"> 自由泊松分布Free Poisson distribution</font>'''作为重复自由卷积的极限在自由概率论中出现
带有跳跃大小 < math > alpha </math > 和速率 < math > lambda </math > 的''' 自由泊松分布Free Poisson distribution'''作为重复自由卷积的极限在自由概率论中出现
This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.
This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.
这个定义类似于从(经典)泊松过程获得经典'''<font color="#ff8000"> 泊松分佈Poisson distribution.</font>'''的一种方法。
这个定义类似于从(经典)泊松过程获得经典''' 泊松分布Poisson distribution.'''的一种方法。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“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 ”
The measure associated to the free Poisson law is given by<ref>James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.</ref>
The measure associated to the free Poisson law is given by<ref>James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.</ref>
Given a sample of n measured values <math> k_i \in \{0,1,...\}</math>, for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is
Given a sample of n measured values <math> k_i \in \{0,1,...\}</math>, for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is
给定一个 n 个测量值的样本{0,1，... } </math > ，对于 i = 1，... ，n，我们希望估计取样的泊松总体参数的值。'''<font color="#ff8000"> 最大似然估计The maximum likelihood estimate</font>'''是
给定一个 n 个测量值的样本{0,1，... } </math > ，对于 i = 1，... ，n，我们希望估计取样的泊松总体参数的值。''' 最大似然估计The maximum likelihood estimate'''是
Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB). Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.
Since each observation has expectation λ so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB). Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.
因为每个观测值都有期望值，所以样本的意义也是如此。因此，'''<font color="#ff8000"> 最大似然估计The maximum likelihood estimate</font>'''是。由于其方差达到了 CRLB 下界，因此它也是一个有效的估计量。它是最小方差无偏的。也可以证明和(因此样本平均值是和的单射)是一个完整充分的统计量。
因为每个观测值都有期望值，所以样本的意义也是如此。因此，''' 最大似然估计The maximum likelihood estimate'''是。由于其方差达到了 CRLB 下界，因此它也是一个有效的估计量。它是最小方差无偏的。也可以证明和(因此样本平均值是和的单射)是一个完整充分的统计量。
To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample <math>\mathbf{x}</math> (called <math>h(\mathbf{x})</math>) and one that depends on the parameter <math>\lambda</math> and the sample <math>\mathbf{x}</math> only through the function <math>T(\mathbf{x})</math>. Then <math>T(\mathbf{x})</math> is a sufficient statistic for <math>\lambda</math>.
To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample <math>\mathbf{x}</math> (called <math>h(\mathbf{x})</math>) and one that depends on the parameter <math>\lambda</math> and the sample <math>\mathbf{x}</math> only through the function <math>T(\mathbf{x})</math>. Then <math>T(\mathbf{x})</math> is a sufficient statistic for <math>\lambda</math>.
为了证明充分性，我们可以用'''<font color="#ff8000"> 因子分解定理Factorization theorem</font>'''。考虑将'''<font color="#ff8000"> 联合泊松分布Joint Poisson distribution</font>'''的'''<font color="#ff8000"> 概率质量函数Probability mass function</font>'''分成两部分: 一部分仅依赖于样本 < math > mathbf { x } </math > (称为 < math > h (mathbf { x }) </math >) ，另一部分依赖于参数 < math > lambda </math > 和样本 < math > mathbf { x } </math > 只通过函数 math < t (mathbf { x }) </math > 。那么 < math > t (mathbf { x }) </math > 就是 < math > lambda </math > 的一个充分的统计量。
为了证明充分性，我们可以用''' 因子分解定理Factorization theorem'''。考虑将''' 联合泊松分布Joint Poisson distribution'''的''' 概率质量函数Probability mass function'''分成两部分: 一部分仅依赖于样本 < math > mathbf { x } </math > (称为 < math > h (mathbf { x }) </math >) ，另一部分依赖于参数 < math > lambda </math > 和样本 < math > mathbf { x } </math > 只通过函数 math < t (mathbf { x }) </math > 。那么 < math > t (mathbf { x }) </math > 就是 < math > lambda </math > 的一个充分的统计量。
To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:
To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:
为了找到泊松族群概率密度函数最大的参数λ ，我们可以使用'''<font color="#ff8000"> 似然函数Likelihood function</font>'''的对数:
为了找到泊松族群概率密度函数最大的参数λ ，我们可以使用''' 似然函数Likelihood function'''的对数:
Solving for λ gives a stationary point.
Solving for λ gives a stationary point.
解出 λ得到'''<font color="#ff8000"> 驻点Stationary point</font>'''。
解出 λ得到''' 驻点Stationary point'''。
which is the negative of n times the reciprocal of the average of the k<sub>i</sub>. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.
which is the negative of n times the reciprocal of the average of the k<sub>i</sub>. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.
它是 n 乘以 k < sub > i </sub > 平均值的倒数。当平均数为正时，这个表达式是负的。如果这一点得到了满足，那么'''<font color="#ff8000"> 驻点The stationary point</font>'''最大化了概率密度函数。
它是 n 乘以 k < sub > i </sub > 平均值的倒数。当平均数为正时，这个表达式是负的。如果这一点得到了满足，那么''' 驻点The stationary point'''最大化了概率密度函数。
=== Confidence interval '''<font color="#ff8000">置信区间Confidence interval </font>'''===
=== Confidence interval '''置信区间Confidence interval '''===
The [[confidence interval]] for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and [[chi-squared distribution]]s. The chi-squared distribution is itself closely related to the [[gamma distribution]], and this leads to an alternative expression. Given an observation ''k'' from a Poisson distribution with mean ''μ'', a confidence interval for ''μ'' with confidence level {{math|1 – α}} is
The [[confidence interval]] for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and [[chi-squared distribution]]s. The chi-squared distribution is itself closely related to the [[gamma distribution]], and this leads to an alternative expression. Given an observation ''k'' from a Poisson distribution with mean ''μ'', a confidence interval for ''μ'' with confidence level {{math|1 – α}} is
The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level is
The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level is
'''<font color="#ff8000">置信区间Confidence interval </font>'''的平均'''<font color="#ff8000"> 泊松分布Poisson distribution</font>'''可以用泊松分布和卡方分布的累积分布函数之间的关系来表示。卡方分布本身与伽玛分布密切相关，这导致了另一种表达方式。给定一个来自平均泊松分佈的观测值 k，一个带有置信水平的置信区间是
'''置信区间Confidence interval '''的平均''' 泊松分布Poisson distribution'''可以用泊松分布和卡方分布的累积分布函数之间的关系来表示。卡方分布本身与伽玛分布密切相关，这导致了另一种表达方式。给定一个来自平均泊松分布的观测值 k，一个带有置信水平的置信区间是
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“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 ”
:<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math>
:<math>\tfrac 12\chi^{2}(\alpha/2; 2k) \le \mu \le \tfrac 12 \chi^{2}(1-\alpha/2; 2k+2), </math>
For application of these formulae in the same context as above (given a sample of n measured values k<sub>i</sub> each drawn from a Poisson distribution with mean λ), one would set
For application of these formulae in the same context as above (given a sample of n measured values k<sub>i</sub> each drawn from a Poisson distribution with mean λ), one would set
为了在与上述相同的上下文中应用这些公式(给定一个 n 个测量值 k < sub > i </sub > 每个取自一个泊松分佈的平均值) ，我们将设置
为了在与上述相同的上下文中应用这些公式(给定一个 n 个测量值 k < sub > i </sub > 每个取自一个泊松分布的平均值) ，我们将设置
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“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 ”
:<math>k=\sum_{i=1}^n k_i ,\!</math>
:<math>k=\sum_{i=1}^n k_i ,\!</math>
=== '''<font color="#ff8000"> Bayesian inference 贝叶斯推理</font>'''===
=== ''' Bayesian inference 贝叶斯推理'''===
In [[Bayesian inference]], the [[conjugate prior]] for the rate parameter ''λ'' of the Poisson distribution is the [[gamma distribution]].{{r|Fink1976}} Let
In [[Bayesian inference]], the [[conjugate prior]] for the rate parameter ''λ'' of the Poisson distribution is the [[gamma distribution]].{{r|Fink1976}} Let
In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution. Let
In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution. Let
在'''<font color="#ff8000"> Bayesian inference 贝叶斯推理</font>'''中，泊松分佈的速率参数的'''<font color="#ff8000"> 共轭先验Conjugate prior</font>'''是伽玛分布。让
在''' Bayesian inference 贝叶斯推理'''中，泊松分布的速率参数的''' 共轭先验Conjugate prior'''是伽玛分布。让
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“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 ”
:<math>\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!</math>
:<math>\lambda \sim \mathrm{Gamma}(\alpha, \beta) \!</math>
The posterior predictive distribution for a single additional observation is a negative binomial distribution, sometimes called a gamma–Poisson distribution.
The posterior predictive distribution for a single additional observation is a negative binomial distribution, sometimes called a gamma–Poisson distribution.
单一额外观察的后验预测分布是负二项分布，有时称为泊松分布。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈 ”改为“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 ”
=== Simultaneous estimation of multiple Poisson means 多重泊松均值的同步估计===
=== Simultaneous estimation of multiple Poisson means 多重泊松均值的同步估计===
Applications of the Poisson distribution can be found in many fields including:
Applications of the Poisson distribution can be found in many fields including:
'''<font color="#ff8000"> 泊松分佈Poisson distribution</font>'''可以应用在很多领域，包括:
''' 泊松分布Poisson distribution'''可以应用在很多领域，包括:
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
* [[Telecommunication]] example: telephone calls arriving in a system.
* [[Telecommunication]] example: telephone calls arriving in a system.
*[[电信]]示例：到达系统的电话。
*[[电信]]示例：到达系统的电话。
The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include:
The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include:
泊松分布过程与泊松过程有关。它适用于各种离散性质的现象(也就是说，那些可能发生0,1,2,3，... 在给定时间内或在给定区域) ，只要现象发生的概率在时间或空间上是常数。可以被模仿为泊松分布的活动包括:
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
<!--
<!--
Comparison of the Poisson distribution (black lines) and the [[binomial distribution with n = 10 (red circles), n = 20 (blue circles), n = 1000 (green circles). All distributions have a mean of 5. The horizontal axis shows the number of events k. As n gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.]]
Comparison of the Poisson distribution (black lines) and the [[binomial distribution with n = 10 (red circles), n = 20 (blue circles), n = 1000 (green circles). All distributions have a mean of 5. The horizontal axis shows the number of events k. As n gets larger, the Poisson distribution becomes an increasingly better approximation for the binomial distribution with the same mean.]]
泊松分布(黑线)与[二项分布(红圈) ，n = 20(蓝圈) ，n = 1000(绿圈)的比较。所有分布的平均值都是5。水平轴显示事件的数量 k。随着 n 变得越来越大，泊松分布变成了一个越来越好的平均二项分布。]
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a [[Bernoulli trial]], where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a [[Bernoulli trial]], where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a Bernoulli trial, where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be <math>\lambda</math>. Divide the whole interval into <math>n</math> subintervals <math>I_1,\dots,I_n</math> of equal size, such that <math>n</math> > <math>\lambda</math> (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval <math>I_i</math> for each <math>i</math> is equal to <math>\lambda/n</math>. Now we assume that the occurrence of an event in the whole interval can be seen as a Bernoulli trial, where the <math>i^{th}</math> trial corresponds to looking whether an event happens at the subinterval <math>I_i</math> with probability <math>\lambda/n</math>. The expected number of total events in <math>n</math> such trials would be <math>\lambda</math>, the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form <math>\textrm{B}(n,\lambda/n)</math>. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as <math>n</math> goes to infinity.
事件的发生率与事件发生在某个小的子间隔(时间、空间或其他)的概率有关。在泊松分佈的例子中，我们假设存在一个足够小的子区间，其中一个事件发生两次的概率是“可以忽略的”。有了这个假设，我们就可以从二项式中推导出泊松分佈，只需要给出整个时间间隔内预期的事件总数的信息。设这个总数是 < math > > lambda </math > 。将整个区间分为 < math > n </math > 子区间 < math > i _ 1，点，i _ n </math > 大小相等，这样 < math > n </math > < math > lambda </math > (因为我们只对区间的很小一部分感兴趣，所以这个假设是有意义的)。这意味着每个 < math > i </math > 中期望的事件数等于 < math > lambda/n </math > 。现在，我们假设一个事件在整个时间间隔内的发生可以被看作是伯努利试验，其中，“ math”试验对应于观察一个事件是否在子时间间隔内发生。在 < math > n </math > 这样的试验中预期的总事件数是 < math > lambda </math > ，这是整个间隔中预期的总事件数。因此，对于区间的每一个细分，我们都近似地将事件的发生作为形式 < math > textrm { b }(n，lambda/n) </math > 的伯努利过程。正如我们之前指出的，我们只想考虑非常小的子区间。因此，我们将极限取为 < math > n </math > 到无穷大。
事件的发生率与事件发生在某个小的子间隔(时间、空间或其他)的概率有关。在泊松分布的例子中，我们假设存在一个足够小的子区间，其中一个事件发生两次的概率是“可以忽略的”。有了这个假设，我们就可以从二项式中推导出泊松分布，只需要给出整个时间间隔内预期的事件总数的信息。设这个总数是 < math > > lambda </math > 。将整个区间分为 < math > n </math > 子区间 < math > i _ 1，点，i _ n </math > 大小相等，这样 < math > n </math > < math > lambda </math > (因为我们只对区间的很小一部分感兴趣，所以这个假设是有意义的)。这意味着每个 < math > i </math > 中期望的事件数等于 < math > lambda/n </math > 。现在，我们假设一个事件在整个时间间隔内的发生可以被看作是伯努利试验，其中，“ math”试验对应于观察一个事件是否在子时间间隔内发生。在 < math > n </math > 这样的试验中预期的总事件数是 < math > lambda </math > ，这是整个间隔中预期的总事件数。因此，对于区间的每一个细分，我们都近似地将事件的发生作为形式 < math > textrm { b }(n，lambda/n) </math > 的伯努利过程。正如我们之前指出的，我们只想考虑非常小的子区间。因此，我们将极限取为 < math > n </math > 到无穷大。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
In this case the binomial distribution converges to what is known as the Poisson distribution by the [[Poisson limit theorem]].
In this case the binomial distribution converges to what is known as the Poisson distribution by the [[Poisson limit theorem]].
In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.
In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.
在这种情况下，二项分布收敛于泊松极限定理所称的泊松分布。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
In such cases n is very large and p is very small (and so the expectation np is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution
In such cases n is very large and p is very small (and so the expectation np is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution
在这种情况下，n 是非常大的，p 是非常小的(所以期望 np 是中等大小)。然后，分布可以近似于不那么麻烦的泊松分佈
在这种情况下，n 是非常大的，p 是非常小的(所以期望 np 是中等大小)。然后，分布可以近似于不那么麻烦的泊松分布
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
:<math>X \sim \textrm{Pois}(np). \,</math>
:<math>X \sim \textrm{Pois}(np). \,</math>
This approximation is sometimes known as the law of rare events,since each of the n individual Bernoulli events rarely occurs. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter np is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.
This approximation is sometimes known as the law of rare events,since each of the n individual Bernoulli events rarely occurs. The name may be misleading because the total count of success events in a Poisson process need not be rare if the parameter np is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour.
这种近似有时被称为稀有事件定律，因为 n 个伯努利事件中的每一个很少发生。这个名称可能有误导性，因为如果参数 np 不小，那么 Poisson 过程中成功事件的总计数就不会很少。例如，一个小时内打给忙碌总机的电话数量跟随着一个泊松分佈，这些事件在接线员看来是频繁的，但是从普通人的角度来看，这些事件很少发生，因为他们不太可能在那个小时内打电话给总机。
这种近似有时被称为稀有事件定律，因为 n 个伯努利事件中的每一个很少发生。这个名称可能有误导性，因为如果参数 np 不小，那么 Poisson 过程中成功事件的总计数就不会很少。例如，一个小时内打给忙碌总机的电话数量跟随着一个泊松分布，这些事件在接线员看来是频繁的，但是从普通人的角度来看，这些事件很少发生，因为他们不太可能在那个小时内打电话给总机。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.
The word law is sometimes used as a synonym of probability distribution, and convergence in law means convergence in distribution. Accordingly, the Poisson distribution is sometimes called the "law of small numbers" because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The Law of Small Numbers is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898.
法律一词有时被用作概率分布的同义词，法律的趋同意味着分配的趋同。因此，泊松分布有时被称为“小数定律” ，因为它是一个事件发生次数的概率分布，这个事件很少发生，但却有很多机会发生。小数定律》是拉迪斯劳斯·博特基威茨的一本关于泊松分布的书，出版于1898年。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
==='''<font color="#ff8000"> Poisson point process 泊松点过程</font>'''===
===''' Poisson point process 泊松点过程'''===
The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space R<sup>d</sup>, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if denotes the number of points in D, then
The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if D is some region space, for example Euclidean space R<sup>d</sup>, for which |D|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if denotes the number of points in D, then
泊松分布是位于某个有限区域的泊松过程的点数。更具体地说，如果 d 是某个区域空间，例如欧几里德空间 r < sup > d </sup > ，对于这个区域 | d | ，区域的面积、体积或者更一般地说，区域的勒贝格测度是有限的，如果表示 d 中的点数，那么
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
=== '''<font color="#ff8000"> Poisson regression and negative binomial regression 泊松回归与负二项回归</font>'''===
=== ''' Poisson regression and negative binomial regression 泊松回归与负二项回归'''===
Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.
Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count (0, 1, 2, ...) of the number of events or occurrences in an interval.
'''<font color="#ff8000"> Poisson regression and negative binomial regression 泊松回归与负二项回归</font>'''分析是有用的，其中依赖(响应)变量是计数(0,1,2，...)的在一个区间内事件发生的数量。
''' Poisson regression and negative binomial regression 泊松回归与负二项回归'''分析是有用的，其中依赖(响应)变量是计数(0,1,2，...)的在一个区间内事件发生的数量。
=== Other applications in science 科学上的其他应用===
=== Other applications in science 科学上的其他应用===
In Causal Set theory the discrete elements of spacetime follow a Poisson distribution in the volume.
In Causal Set theory the discrete elements of spacetime follow a Poisson distribution in the volume.
在'''<font color="#ff8000"> 因果集合论Causal Set theory</font>'''中，时空的离散元素在集合中遵循一个泊松分佈。
在''' 因果集合论Causal Set theory'''中，时空的离散元素在集合中遵循一个泊松分布。
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分佈”改为“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”
==Computational methods计算方法==
==Computational methods计算方法==
The Poisson distribution poses two different tasks for dedicated software libraries: Evaluating the distribution <math>P(k;\lambda)</math>, and drawing random numbers according to that distribution.
The Poisson distribution poses two different tasks for dedicated software libraries: Evaluating the distribution <math>P(k;\lambda)</math>, and drawing random numbers according to that distribution.
泊松分布为专用软件库提出了两个不同的任务: 评估分布 < math > p (k; lambda) </math > ，并根据分布绘制随机数。
=== '''<font color="#ff8000"> Evaluating the Poisson distribution 计算泊松分布</font>'''===
=== ''' Evaluating the Poisson distribution 计算泊松分布'''===
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
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，传统上对泊松分佈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
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】“泊松分佈”改为“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”
:<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>
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'''<font color="#ff8000"> 内置函数Built-in functions</font>'''来评估泊松分佈
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】“泊松分佈”改为“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”
* [[R (programming language)|R]]: function <code>dpois(x, lambda)</code>;
* [[R (programming language)|R]]: function <code>dpois(x, lambda)</code>;
The less trivial task is to draw random integers from the Poisson distribution with given <math>\lambda</math>.
The less trivial task is to draw random integers from the Poisson distribution with given <math>\lambda</math>.
更简单的任务是用给定的 < math > lambda </math > 从泊松分佈中提取随机整数。
更简单的任务是用给定的 < math > lambda </math > 从泊松分布中提取随机整数。
=== Generating Poisson-distributed random variables 生成泊松分布随机变量===
===生成泊松分布随机变量===
给出了一个产生随机泊松分布数(伪随机数抽样)的简单算法:
算法：泊松随机数 Knuth :
:初始化:
::设 L ← ''e''<sup>−λ</sup>, k ← 0 且 p ← 1.
:执行
:: k ← k + 1.
::在[0,1]中生成均匀随机数 u 并且设 p ← p × u。
:而 p > L.
:返回 k − 1.
返回值''k''的复杂度是线性的，平均为λ。还有许多其他的算法可以改进这一点。一些是在 Ahrens & Dieter，见下面。
For large values of λ, the value of L = ''e''<sup>−λ</sup> may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that ''e''<sup>−STEP</sup> does not underflow: {{Citation needed|reason=Original source is missing|date=March 2019}}
For large values of λ, the value of L = e<sup>−λ</sup> may be so small that it is hard to represent. This can be solved by a change to the algorithm which uses an additional parameter STEP such that e<sup>−STEP</sup> does not underflow:
对于较大的值，l = e </sup >-</sup > 的值可能非常小，以至于很难表示。这可以通过改变算法来解决，该算法使用附加参数 STEP，使得 e < sup >-STEP </sup > 不会底流:
--[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“底流”改为“下溢”
'''algorithm''' ''poisson random number (Knuth)'':
'''algorithm''' ''poisson random number (Junhao, based on Knuth)'':
algorithm poisson random number (Knuth):
algorithm poisson random number (Junhao, based on Knuth):
泊松随机数算法(Junhao，基于 Knuth) :
'''init''':
'''init''':
初始化:
初始化:
'''Let''' L ← ''e''<sup>−λ</sup>, k ← 0 and p ← 1.
'''Let''' λLeft ← λ, k ← 0 and p ← 1.
Let L ← e<sup>−λ</sup>, k ← 0 and p ← 1.
Let λLeft ← λ, k ← 0 and p ← 1.
Let L ← e<sup>−λ</sup>, k ← 0 and p ← 1.
Let λLeft ← λ, k ← 0 and p ← 1.
'''do''':
'''do''':
k ← k + 1.
k ← k + 1.
Generate uniform random number u in [0,1] and '''let''' p ← p × u.
Generate uniform random number u in (0,1) and '''let''' p ← p × u.
Generate uniform random number u in [0,1] and let p ← p × u.
Generate uniform random number u in (0,1) and let p ← p × u.
在[0,1]中生成均匀随机数 u 并且设 p ← p u。
在(0,1)中生成均匀随机数 u 并且设 p ← p u。
'''while''' p < 1 and λLeft > 0:
while p < 1 and λLeft > 0:
while p < 1 and λLeft > 0:
'''if''' λLeft > STEP:
if λLeft > STEP:
if λLeft > STEP:
p ← p × ''e''<sup>STEP</sup>
p ← p × e<sup>STEP</sup>
p ← p × e<sup>STEP</sup>
λLeft ← λLeft − STEP
λLeft ← λLeft − STEP
λLeft ← λLeft − STEP
'''else''':
else:
其他:
p ← p × ''e''<sup>λLeft</sup>
p ← p × e<sup>λLeft</sup>
p ← p × e<sup>λLeft</sup>
λLeft ← 0
λLeft ← 0
λLeft ← 0
'''while''' p > 1.
while p > 1.
同时 p > 1。
'''return''' k − 1.
return k − 1.
return k − 1.
The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near ''e''<sup>700</sup>, so 500 shall be a safe ''STEP''.
The choice of STEP depends on the threshold of overflow. For double precision floating point format, the threshold is near e<sup>700</sup>, so 500 shall be a safe STEP.
STEP 的选择取决于溢出''' 阈值Threshold'''。对于双精度浮点格式，''' 阈值Threshold'''接近 e < sup > 700 </sup > ，因此500应该是一个安全的 STEP。
Other solutions for large values of λ include [[rejection sampling]] and using Gaussian approximation.
Other solutions for large values of λ include rejection sampling and using Gaussian approximation.
其他λ的大数值的求解包括抑制取样和使用''' 高斯近似Gaussian approximation'''。
[[Inverse transform sampling]] is simple and efficient for small values of λ, and requires only one uniform random number ''u'' per sample. Cumulative probabilities are examined in turn until one exceeds ''u''.
Inverse transform sampling is simple and efficient for small values of λ, and requires only one uniform random number u per sample. Cumulative probabilities are examined in turn until one exceeds u.
逆变换采样对于λ小数值的样本是简单有效的，并且每个样本只需要一个均匀一致的随机数 u。依次检查累积概率，直到其超过 u。
'''algorithm''' ''Poisson generator based upon the inversion by sequential search'':{{r|Devroye1986|p=505}}
algorithm Poisson generator based upon the inversion by sequential search:
基于顺序检索反演的''' 泊松发生器算法Algorithm Poisson generator ''':
'''init''':
init:
初始化:
'''Let''' x ← 0, p ← ''e''<sup>−λ</sup>, s ← p.
Let x ← 0, p ← e<sup>−λ</sup>, s ← p.
Let x ← 0, p ← e<sup>−λ</sup>, s ← p.
Generate uniform random number u in [0,1].
Generate uniform random number u in [0,1].
在[0,1]中生成均匀随机数 u。
'''while''' p > 1.
'''while''' u > s '''do''':
while p > 1.
while u > s do:
做以下步骤:
x ← x + 1.
x ← x + 1.
x ← x + 1.
p ← p × λ / x.
p ← p × λ / x.
p ← p × λ / x.
s ← s + p.
s ← s + p.
s ← s + p.
'''return''' x.
return x.
返回 x。
== 历史 ==
这种分布最早由西蒙·丹尼斯·泊松 Siméon Denis Poisson（1781-1840）提出，并与他的概率论一起发表在他的著作Recherches sur la probabilité des jugements en matière criminelle et en matière Civile (1837) 中。这项工作通过关注某些随机变量''N''，其中包括在给定时间间隔内发生的离散事件（有时称为“事件”或“到达事件”）的数量来推断某一国家的错误定罪数量。这个结果早在1711年就已由亚伯拉罕·德·莫弗 Abraham de Moivre给出了。这使它成为'''斯蒂格勒定律 Stigler's law'''的一个例子，也使一些作者提出，泊松分布应该以de Moivre的名字命名。
1860年，Simon Newcomb 将泊松分布与天文台一个空间单位中发现的恒星数量进行了比较。
这种分布的进一步实际应用是在1898年，当时拉迪斯劳斯·博特基威茨被赋予任务调查普鲁士军队中被马踢意外杀死的士兵人数; 这个实验将泊松分布引入可靠性技术领域。
==参考文献==
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[[Category:泊松分布]]
[[Category:共轭先验分布]]
[[Category:阶乘和二项式主题]]
[[Category:无限可分的概率分布]]
[[Category:Poisson distribution| ]]
[[Category:Articles with example pseudocode]]
[[Category:Conjugate prior distributions]]
[[Category:Factorial and binomial topics]]