故事 · 被马踢死的普鲁士士兵 Origin Story · Prussian Soldiers Killed by Horse-Kicks
泊松分布专数"单位时间或空间里,稀有事件发生几次":每小时来几通电话、每页书有几个错别字、每平方米布料几个瑕疵。 它最传奇的应用,来自统计学家 Ladislaus Bortkiewicz:他收集了普鲁士 14 个骑兵军团 20 年间 被马踢死的士兵人数,发现这种又罕见又随机的事件,竟服服帖帖落在泊松曲线上。 泊松只有一个参数 λ,而且 λ 既是均值也是方差 —— 这是它最好认的指纹。 λ 小时分布右偏、拖一条小尾巴;λ 大时柱群越来越对称,趋近正态钟形。 Poisson counts how many rare events happen in a unit of time or space: calls per hour, typos per page, defects per square metre of fabric. Its most famous demonstration came from statistician Ladislaus Bortkiewicz, who tallied soldiers kicked to death by horses across 14 Prussian cavalry corps over 20 years — and found that this rare, random tragedy landed obediently on a Poisson curve. First named in 1837 by Siméon Denis Poisson, the distribution has just one parameter, λ, which is both the mean and the variance — its most recognisable fingerprint. Small λ gives a right-skewed shape with a long tail; large λ pulls the bars into a symmetric bell that looks essentially normal.

1 这一天 24 小时,病人随机到达 A 24-Hour Day of Random Arrivals

每格 = 1 小时1 cell = 1 hour

2 "这一小时来 k 个"的概率分布 P(k arrivals this hour)

右偏right-skewed

紫柱=泊松理论概率 P(X=k);红点=刚才那天 24 小时的实际频率。点数越多,红点越贴合紫柱。 Purple bars = the theoretical Poisson P(X=k); red dots = the empirical frequency from the simulated day. More samples → red dots hug the purple bars tighter.

3 现实里的泊松 Poisson in the Real World

缺陷率:每件产品、每米焊缝的缺陷数服从泊松,是 c 控制图、u 控制图的理论基础。 Defect rate: defects per unit or per metre of weld follow Poisson — the theoretical engine behind the c-chart and u-chart.
到达计数:单位时间的来电、网站请求、急诊到达数,是排队论与容量规划的输入。 Arrival counts: calls per hour, web requests per second, ER walk-ins — the raw input to queuing theory and capacity planning.
DPMO 基础:每百万机会的缺陷数(DPMO)、西格玛水平的计算,都把缺陷视为泊松事件。 Basis of DPMO: defects per million opportunities (DPMO) and the sigma-level conversion both treat defects as Poisson events.
排队论:泊松到达 + 指数服务(下一页)组成 M/M/1 等经典排队模型,配座席、定产能全靠它。 Queuing theory: Poisson arrivals plus exponential service (next page) build the classic M/M/1 model — staff sizing and capacity targets all ride on it.
一句话In One Line
泊松只需一个参数 λ(均值=方差=λ),就能描述一切"稀有独立事件的计数":呼叫中心话务、产线缺陷数、保险索赔、服务器请求。 它是排队论、可靠性、缺陷管理(DPU/DPMO)的地基。λ 越大,泊松越对称,λ>20 时几乎就是正态 —— 又一次中心极限的体现。 One parameter — λ (mean = variance) — is all Poisson needs to describe any count of "rare independent events": call-centre traffic, line defects, insurance claims, server requests. It is the foundation of queuing theory, reliability work, and defect management (DPU/DPMO). As λ grows the shape symmetrises, and by λ > 20 it is essentially a normal bell — the central limit theorem showing up once again.
常见误用Common Mistakes
事件不独立还用泊松(如传染、扎堆促销)。泊松要求事件相互独立、发生率恒定。 Using Poisson when events are not independent (contagion, promo spikes). Poisson requires independent events at a constant rate.
方差远大于均值仍套泊松。过度离散时改用负二项分布。 Forcing Poisson when variance ≫ mean. For overdispersed counts, switch to the negative binomial.
把"平均 λ"当成每小时必来 λ 个。λ 只是长期均值,单小时波动很大。 Treating "λ on average" as a guaranteed λ every hour. λ is only the long-run mean; single-hour counts can swing wildly.

泊松分布