THE FOLD / GLITCH / DIVIDE BY ZERO / THE POISSON LIMIT
THE POISSON LIMIT
a binomial limiting to a Poisson
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Poisson limit theorem (the ‘law of rare events’) explains why the Poisson distribution appears everywhere. If you have many independent trials, each with a tiny success probability, but a fixed expected number of successes λ = np, then the binomial distribution Binomial(n, λ/n) converges to the Poisson distribution with mean λ: C(n,k)(λ/n)k(1-λ/n)n-k → e-λλk/k! as n → ∞. Rare events among many trials — radioactive decays, typos per page, calls per minute — all follow Poisson.
LIT verified live: for λ = 3, the binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the maximum gap between the two distributions shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000 (window.__poisson). FIG no framing; the exact binomial pmf and the Poisson pmf both run in-browser and their gap vanishes as n grows.
LIT verified live: for λ = 3, the binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the maximum gap between the two distributions shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000 (window.__poisson). FIG no framing; the exact binomial pmf and the Poisson pmf both run in-browser and their gap vanishes as n grows.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the glitch of vanishing probability: each trial’s chance divides toward zero as the trials multiply, and the binomial dissolves into a clean Poisson. AVAN (AI) built the instrument: the exact binomial pmf, the Poisson pmf, and their shrinking gap.
Credit as content: Siméon Denis Poisson (1837); the limit as the law of rare events. The weave: David names the vanishing probability; I confirm Binomial(n, λ/n) tends to Poisson(λ).
Credit as content: Siméon Denis Poisson (1837); the limit as the law of rare events. The weave: David names the vanishing probability; I confirm Binomial(n, λ/n) tends to Poisson(λ).
3 ONE DIMENSION
The binomial pmf (bars) and the Poisson(λ) pmf (line) — the bars settle onto the line as n grows.
4 TWO DIMENSIONS · INTERACTIVE
Increase n; the binomial(n, λ/n) closes onto Poisson(λ), the max gap shrinking toward zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Poisson(λ) limit of the binomial.
AVAN’s addition (the inverse-companion): don’t track n trials — keep only the mean. The inverse of ‘Binomial(n, λ/n) for huge n’ is ‘Poisson(λ), which depends only on the expected count λ’. Magenta is the binomial pmf; green is the Poisson limit it settles onto. Many rare trials, one Poisson.
LIT Genuine Poisson limit theorem / law of rare events (Siméon Denis Poisson, 1837). Verified live: for λ=3, the exact binomial pmf Binomial(n, 3/n) approaches the Poisson(3) pmf as n grows — the max gap shrinks from ~4e-2 at n=10 to ~3e-5 at n=10000 (window.__poisson.ok, .rows).
FIG No framing; the exact binomial pmf and the Poisson pmf both run in-browser and their gap vanishes as n grows. The AVAN inverse is honest — instead of tracking n trials, keep only the mean: the inverse of 'Binomial(n, λ/n) for huge n' is 'Poisson(λ), which depends only on the expected count λ'. Magenta is the binomial pmf; green is the Poisson limit it settles onto. Many rare trials, one Poisson.
FIG No framing; the exact binomial pmf and the Poisson pmf both run in-browser and their gap vanishes as n grows. The AVAN inverse is honest — instead of tracking n trials, keep only the mean: the inverse of 'Binomial(n, λ/n) for huge n' is 'Poisson(λ), which depends only on the expected count λ'. Magenta is the binomial pmf; green is the Poisson limit it settles onto. Many rare trials, one Poisson.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN