Poisson distribution

The formula

The probability of seeing exactly k events when the average is λ:

P(X = k) = λ^k · e^(−λ) / k!        k = 0, 1, 2, ...

Reading: the whole distribution runs on a single knob. Set the expected count λ and every probability follows — no separate spread parameter to estimate, because there isn't one.

The average comes from a rate times an exposure:

λ = r · T      (rate per unit exposure × amount of exposure)

Reading: 0.4 failures per operating year watched for 30 months is λ = 1.0. Double the boards inspected, double λ. Counts only compare across equal exposures.

The moments, which are the signature of the distribution:

mean = λ       variance = λ       σ = √λ

Reading: the spread is not free to vary — it is chained to the mean. Data whose variance runs well above its mean is telling you it isn't Poisson.

The two workhorse special cases:

P(0) = e^(−λ)                        (probability of a clean sheet)
P(≥1) = 1 − e^(−λ)                   (probability of at least one)

Reading: at λ = 2 defects per board, 13.5% of boards come out clean and 5.3% carry five or more. Both numbers fall straight out of the one-knob formula.

And the additivity rule:

X₁ ~ Poisson(λ₁), X₂ ~ Poisson(λ₂), independent  ⇒  X₁ + X₂ ~ Poisson(λ₁ + λ₂)

Reading: counts pool. Ten test stands each logging their own rare failures behave, combined, like one stand with the summed rate — which is why fleet-level data is worth so much more than unit-level data.

Where you meet it

On the production line, at the c-chart. Defects per board is the textbook case: thousands of solder joints (many opportunities), each with a small chance of a flaw. The control limits on a count chart sit at λ ± 3√λ because the Poisson ties the spread to the mean — nobody estimated a standard deviation, the distribution supplied it.
At the reliability scoreboard. A fleet running at a constant failure rate r produces Poisson-distributed failure counts in any window: λ = r·T over the pooled operating hours. This is the machinery behind zero-failure demonstration tests — run long enough that λ would be around 3 if the disputed rate were real, see nothing, and you've bounded the rate (next section).
At the radar console and the photon counter. False alarms per scan, photoelectrons per integration window, counts per second on a radiation survey meter — all Poisson, because each arrival is an independent rare event. The √N counting error every instrumentation engineer carries around is the Poisson standard deviation wearing work clothes: collect 100 counts and your relative uncertainty is √100/100 = 10%, no matter how good the detector is.
At the review board, arguing whether a number moved. Three anomalies last quarter, seven this quarter. Before anyone reorganizes anything: a Poisson count of 3 has σ ≈ 1.7 and a count of 7 has σ ≈ 2.6, and a jump of that size out of pure randomness is not remarkable. The Poisson is the null hypothesis for "did the count actually change."

How it works

The engine is the law of rare events. Take a binomial — n independent opportunities, each with small probability p — and let n grow while p shrinks with the product n·p = λ held fixed. The binomial collapses onto the Poisson ^[1]^[2]. That limit is why the same distribution covers solder joints, alpha particles, and phone calls: the physics differs, the structure — many chances, each unlikely, none influencing the others — is identical. The approximation is aggressive: at n = 1000, p = 0.002, the binomial gives P(0) = 0.1351 against the Poisson's 0.1353, and P(3) = 0.1806 against 0.1804. Verified numerically. In practice you never know n or p for real hardware; the Poisson lets you skip both and estimate the one thing you can measure, the average count.

The same object viewed from the time axis is the Poisson process: if counts in any window are Poisson, the waiting times between events are exponential and memoryless. A system that has run 900 hours since the last failure is no more "due" than one that failed yesterday — under this model, the hazard never ages. That is a modeling assumption about constant rates, not a law of nature, and it is the assumption you are silently making every time you quote MTBF from pooled field counts.

The gotchas, in the order they bite:

Variance equals mean is a claim, not a courtesy. Real count data frequently runs overdispersed — variance well above the mean — because the rate isn't constant: one bad solder-paste lot, one connector vendor, one rainy test week. Clustered causes produce clustered counts, and Poisson limits drawn on overdispersed data throw false alarms constantly. Check the ratio s²/x̄ before trusting the model; if it's far above 1, you need a negative binomial or, better, the assignable cause. (Mild rate variation is survivable — Bortkiewicz's horse-kick data pooled fourteen different corps and still fit, mean 0.70 against variance 0.76 ^[5]^[6] — but batch-correlated events are not.)

Exposure normalization. λ scales with the window. Comparing this month's count to last month's without checking that operating hours, boards built, or scans run were equal is the most common way count data lies. Failure counts divide by exposure before they mean anything.

Zero observed is not zero rate. No failures in the test window says the rate is probably small, not that it's zero. The exact statement: with zero events observed over exposure T, the 95% upper confidence bound on the rate is λ ≤ 3.0 (precisely −ln 0.05 = 2.996), i.e. r ≤ 3/T. That "rule of three" is the honest answer to "we tested and saw nothing" — and the reason the demonstration test's length, not its cleanliness, determines what it proves.

Small counts are loud. Relative uncertainty is 1/√λ. A count of 4 carries ±50%. Decisions hung on single-digit counts — anomalies, escapes, near misses — are decisions hung on noise unless the change is large or the data is pooled. This is where the additivity rule earns its keep: pool the fleet.

The normal approximation is fine late, ugly early. By λ ≈ 20 a Gaussian with continuity correction is honest — P(X ≤ 15) comes out 0.1572 against the exact 0.1565. Below λ ≈ 10 the Poisson is visibly skewed (skewness 1/√λ ^[1]), the Gaussian bleeds probability below zero, and symmetric ±2σ intervals misstate both tails. Use exact Poisson math for small counts; it's one line of any stats library.

History

The name is a case study in credit arriving late. Abraham de Moivre worked out the limiting form of the binomial for rare events in De Mensura Sortis, published in the Philosophical Transactions in 1711 — more than a century before the man the distribution is named for ^[1]^[4]. Siméon Denis Poisson, the École Polytechnique professor who put his name on half of mathematical physics, reached it in his 1837 treatise Recherches sur la probabilité des jugements en matière criminelle et en matière civile — a book about the probability of jury verdicts, of all things, in which he also coined the phrase "law of large numbers" ^[1]^[3]. For Poisson the distribution was a passing limit result; he died in 1840 with no idea it would carry his name onto every reliability engineer's desk.

The distribution sat as a curiosity until 1898, when Ladislaus Bortkiewicz published Das Gesetz der kleinen Zahlen — the law of small numbers — and showed it fit real data: deaths of Prussian cavalrymen from horse kicks, fourteen corps tracked over twenty years, 196 deaths in 280 corps-years ^[5]^[6]. The fit was good even though the corps plainly differed from one another, which was Bortkiewicz's deeper point: rare-event counts behave Poisson even when the underlying probabilities wobble ^[5]^[6].

Engineers then discovered it independently, twice, within two years. A. K. Erlang at the Copenhagen Telephone Company showed in 1909 that randomly placed telephone calls follow the Poisson distribution, founding traffic engineering on it ^[8]^[11]. In 1910 Rutherford and Geiger counted alpha-particle scintillations in 2,608 intervals of one-eighth of a minute (7.5 seconds) — mean 3.87 per interval — and found the fluctuations exactly Poisson, with Harry Bateman supplying the mathematics in an appendix ^[7]^[11]. Radioactive decay remains the cleanest Poisson source in nature, which is why counting statistics labs still reproduce that experiment.

The most famous engineering application came out of the Blitz. In 1946 the actuary R. D. Clarke took 537 V-1 flying-bomb impacts on south London, gridded the area into 576 quarter-square-kilometer cells, and tested the popular wartime claim that the bombs clustered on targets. The Poisson with λ = 537/576 = 0.932 predicted 226.7 empty cells; London had 229. Every bin matched ^[9]^[10]. The weapons were falling at random — a statistical answer to an intelligence question, computed by hand.

Related tools

/tools/half-life-decay — radioactive decay is the canonical Poisson source: the decay law gives the rate, and counts in any fixed window are Poisson around it
/tools/bearing-life-l10 — the neighboring reliability math: Poisson counts failures when the rate is constant; L10/Weibull models the life when it isn't
/tools/thermal-noise-floor — the other line in the detector noise budget: thermal noise is Gaussian, but the shot-noise term beside it is Poisson counting of discrete electrons

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →