故事 · 被马踢死的普鲁士士兵
Origin Story · Prussian Cavalry and the Horse Kick
1837 年,法国数学家 Siméon Denis Poisson 在研究司法判决中的小概率事件时写下这条公式。半个世纪后,
统计学家 Ladislaus Bortkiewicz 用它做了一件惊艳的事: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,竟精确地落在泊松曲线上。从此泊松成了"稀有事件计数"的代名词 ——
凡是"单位时间/空间里,发生概率很小但机会很多"的事,都归它管。它和二项是亲戚:当二项的 n 很大、p 很小而 np = λ 适中时,二项就坍缩成泊松。
In 1837 the French mathematician Siméon Denis Poisson wrote down the formula while studying low-probability outcomes in judicial verdicts. Half a century later,
statistician Ladislaus Bortkiewicz pulled off a stunning demonstration: he gathered twenty years of records from 14 Prussian cavalry corps tracking
soldiers killed by horse kicks, and found this rare, random event landed almost perfectly on the Poisson curve. From that moment on, Poisson became the
default vocabulary for "counting rare events" — anywhere you have many trials with a tiny individual probability in a fixed window. It is the close cousin of the binomial:
when n is large and p is small with np = λ held moderate, the binomial collapses straight into Poisson.
1 λ 小则右偏稀有,λ 大则长成钟形 Small λ → Right-Skewed and Rare; Large λ → Bell-Shaped
右偏2 二项 n大p小 → 泊松:两条几乎重合 Binomial (large n, small p) → Poisson: The Two Curves Coincide
点按钮叠加click to overlay取 n = 200、p = λ/200(让 np = λ),紫柱是二项、红点连线是泊松。 事件越稀有(n 大 p 小),两者越难分辨 —— 这就是泊松被称作"稀有事件极限"的原因。 Set n = 200 and p = λ/200 so np = λ. Purple bars are the binomial, the red dotted curve is Poisson. The rarer the event (larger n, smaller p), the harder it is to tell them apart — which is exactly why Poisson is called the rare-event limit of the binomial.
3 现实里的泊松分布 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 counts: defects per unit or per meter of weld follow Poisson — the theoretical backbone of the c-chart and u-chart.
到达计数:单位时间来电、网站请求、急诊到达数,排队论与容量规划的输入。
Arrival counts: inbound calls, web requests, ER arrivals per unit time — the input to queueing 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。
DPMO foundation: defects per million opportunities (DPMO) and sigma-level calculations 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。
Queueing theory: Poisson arrivals plus exponential service (next page) give us the M/M/1 and other classic queueing models.
一句话In One Line
泊松数的是"单位时间/空间里,稀有事件发生几次",唯一旋钮 λ 同时是它的均值和方差 ——
这是它最好认的指纹(若实测方差远大于均值,说明数据"过度离散",泊松不再合适)。
它处在分布家族的枢纽位置:往一边走,二项 n 大 p 小趋近它;往另一边走,λ 大时它趋近正态。
再问"下一次事件还要等多久",答案就是它的连续表亲 —— 指数分布。掌握泊松,缺陷率、到达计数、DPMO 与排队论就都通了。
Poisson counts how many rare events happen in a fixed window, and its single dial λ is simultaneously its mean and its variance —
the easiest fingerprint to spot (if your sample variance is much larger than the mean, the data is "over-dispersed" and Poisson no longer fits).
It sits at the crossroads of the distribution family: from one side, the binomial with large n, small p collapses into it; from the other,
large λ bends Poisson toward the normal. Ask "how long until the next event?" and the answer is its continuous cousin, the exponential.
Master Poisson and defect rates, arrival counts, DPMO, and queueing theory all click into place.
常见误用Common Mistakes
方差远大于均值还硬套泊松。出现"过度离散"时改用负二项分布;泊松要求均值≈方差。
Forcing Poisson when variance ≫ mean. When the data is over-dispersed, switch to the negative binomial — Poisson demands mean ≈ variance.
事件并不独立/有聚集还用泊松。故障会扎堆、到达有高峰时独立假设失效,结果会低估尾部。
Applying Poisson when events cluster or aren't independent. Failures bunch up and arrivals spike — the independence assumption breaks and you'll underestimate the tail.
把缺陷数(计数)当不良率(比例)。一件可有多个缺陷用泊松;一件只判合格/不合格用二项。
Confusing defect counts with defective rates. Use Poisson when one unit can carry multiple defects; use the binomial when each unit is just pass/fail.