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第1,787行： − + + + − {{main|Poisson limit theorem}} − − [[File:Binomial versus poisson.svg|right|upright=1.5|thumb |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. − − 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. − --[[用户:fairywang|fairywang]]([[用户讨论:fairywang|讨论]]) 【审校】“泊松分布”改为“泊松分布” − − − In several of the above examples—such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the [[binomial distribution]], that is − − In several of the above examples—such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is − − 在上面的几个例子中---- 例如，一个给定序列的 dna 突变的数量---- 被计算的事件实际上是离散试验的结果，也就是说，更准确地说，是用二项分布模型来模拟的 + − <math>X \sim \textrm{B}(n,p). \,</math> − X sim textrm { b }(n，p).，math+ − − − − 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{{Citation needed|date=April 2012}} − − 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'',{{r|Cameron1998|p=5}}since each of the ''n'' individual [[Bernoulli distribution|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.{{r|vonBortkiewitsch1898}}{{r|Edgeworth1913}} − − 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】“泊松分布”改为“泊松分布” − − − − − − {{Main|Poisson point process}} − − − − 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 {{nowrap|''N''(''D'')}} 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】“泊松分布”改为“泊松分布” − <math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>+ − − < math > p (n (d) = k) = frac {(lambda | d |) ^ k e ^ {-lambda | d | }{ k! }. math − −
→Law of rare events稀有事件定律
=== Law of rare events稀有事件定律 ===
=== 稀有事件定律 ===
[[File:Binomial versus poisson.svg|right|upright=1.5|thumb |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。]]
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,\dots,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</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>i^{th}</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。
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。
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。
在上面的几个例子中---- 例如，一个给定序列的DNA突变的数量---- 被计算的事件实际上是离散试验的结果，也就是说，更准确地说，是用二项分布模型来模拟的
:<math>X \sim \textrm{B}(n,p). \,</math>
:<math>X \sim \textrm{B}(n,p). \,</math>
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
:<math>X \sim \textrm{Pois}(np). \,</math>
:<math>X \sim \textrm{Pois}(np). \,</math>
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。
规律 law一词有时被用作概率分布的同义词，规律的收敛意味着分布的收敛。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 Ladislaus Bortkiewicz的一本关于泊松分布的书，出版于1898年。
===泊松点过程===
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
===''' Poisson point process 泊松点过程'''===
: <math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>
: <math> P(N(D)=k)=\frac{(\lambda|D|)^k e^{-\lambda|D|}}{k!} .</math>
<br>
=== ''' Poisson regression and negative binomial regression 泊松回归与负二项回归'''===
=== ''' Poisson regression and negative binomial regression 泊松回归与负二项回归'''===