The Poisson Distribution: From Prussian Deaths to Website Traffic
In 1898, a statistician counted soldiers kicked to death by horses. The data followed a distribution that now models everything from earthquakes to Amazon server requests.
In 1898, Russian statistician Ladislaus von Bortkiewicz published a study called Das Gesetz der kleinen Zahlen — The Law of Small Numbers. He had collected data on soldiers in 14 Prussian cavalry corps killed by horse kicks, across 20 years. The data was gruesome and meticulous: 196 deaths total, distributed across 280 corps-years of observation.
Von Bortkiewicz fit this data to a probability distribution that Siméon Denis Poisson had described in 1837. The fit was almost perfect. From soldiers trampled by horses to Amazon Web Services handling millions of server requests per second, that same distribution governs an astonishing variety of processes.
When events are Poisson-distributed
The Poisson distribution applies when: events occur randomly and independently, the average rate λ is constant over the observation period, and two events cannot occur at exactly the same instant. Under these conditions, the number of events k in a fixed interval follows:
P(X = k) = (λᵏ × e⁻λ) / k! Where: k = number of events (0, 1, 2, ...) λ = average rate of events per interval e = Euler's number ≈ 2.71828 Mean = λ Variance = λ (equal to the mean — a key property)
Von Bortkiewicz's data
| Deaths per corps-year | Observed frequency | Poisson prediction (λ = 0.70) |
|---|---|---|
| 0 | 144 | 139.0 |
| 1 | 91 | 97.3 |
| 2 | 32 | 34.1 |
| 3 | 11 | 7.9 |
| 4+ | 2 | 1.7 |
The theoretical predictions, calculated from a single parameter λ = 0.70, match the observed data with remarkable precision. The Poisson distribution had found its first famous application.
Modern applications
The Poisson distribution models any process where rare, independent events occur against a time or space background:
| Domain | Event | Typical λ |
|---|---|---|
| Telecommunications | Calls arriving at a switchboard per minute | varies by time of day |
| E-commerce | Orders placed per second during peak | 10–1000 |
| Physics | Radioactive decay events per second | determined by half-life |
| Insurance | Claims per policy per year | 0.05–0.5 |
| Astronomy | Photons hitting a telescope per millisecond | varies by source brightness |
| Biology | Mutations per DNA strand per replication | 10⁻⁹ per base pair |
The connection to the binomial distribution
Binomial: P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ As n → ∞ and p → 0 with np = λ (constant): → Binomial converges to Poisson(λ) Intuition: many trials (n large), rare success (p small), constant expected count (np = λ). This is exactly the structure of "rare events in large populations." Website visits: n = millions of users, p = tiny probability any given user visits in the next second.
Queueing theory and Poisson arrivals
When customers arrive at a queue following a Poisson process with rate λ, and are served with exponential service times with rate μ, the system is called an M/M/1 queue — the simplest model in queueing theory. The average number of customers in the system is λ/(μ−λ), which explodes to infinity as λ approaches μ. This is why systems under heavy load degrade catastrophically rather than gracefully.
Every major cloud provider uses Poisson arrival models to provision capacity. If Amazon Web Services underestimates λ for Prime Day, the queue grows without bound. They do not.