Distribution Lab

Poisson, Pareto, Zipf, Beta, Cauchy — sampled and plotted.

numbers.distribution · v1 · renderer chart

5, 2, 5, 4, 4, 4, 4, 7, 4, 3, 2, 3, 3, 4, 3, 3, 10, 3, 3, 2, 4, 2, 2, 1, 5, 2, 7, 1, 7, 5, 5, 8, 3, 9, 3, 4, 6, 6, 2, 3, 2, 3, 3, 3, 4, 7, 4, 4, 4, 7, 3, 4, 1, 2, 2, 2, 2, 1, 4, 6, 6, 6, 3, 3, 4, 4, 0, 3, 2, 2, 2, 3, 2, 5, 3, 7, 1, 10, 1, 1, 0, 9, 4, 2, 3, 5, 2, 5, 4, 5, 7, 2, 2, 7, 7, 6, 5, 7, 2, 2, 6, 2, 1, 5, 6, 3, 6, 2, 6, 0, 8, 3, 3, 5, 3, 4, 5, 8, 4, 5, 5, 4, 3, 5, 4, 5, 7, 8, 4, 1, 4, 5, 2, 4, 3, 7, 2, 3, 2, 5, 3, 3, 4, 2, 6, 4, 3, 3, 5, 5, 3, 4, 6, 5, 2, 8, 1, 4, 1, 2, 3, 4, 8, 5, 4, 4, 5, 3, 6, 2, 3, 5, 4, 2, 2, 3, 3, 4, 4, 5, 3, 4, 7, 4, 2, 2, 8, 2, 7, 5, 5, 3, 2, 8, 2, 5, 5, 3, 3, 5, 4, 5, 5, 4, 6, 7, 2, 2, 2, 2, 1, 2, 3, 2, 7, 5, 6, 5, 7, 7, 4, 5, 6, 7, 3, 3, 3, 4, 2, 10, 7, 5, 6, 2, 5, 4, 8, 5, 4, 7, 2, 2, 5, 5, 3, 4, 1, 3, 4, 2, 4, 4, 6, 4, 5, 6, 2, 3, 3, 7, 1, 4, 3, 5, 8, 3, 2, 4, 5, 1, 4, 3, 3, 7, 2, 4, 4, 6, 3, 2, 2, 5, 3, 3, 1, 0, 5, 3, 5, 5, 2, 7, 3, 7, 7, 6, 2, 2, 2, 3, 4, 4, 2, 1, 3, 3, 5, 5, 5, 4, 4, 1, 4, 4, 3, 6, 3, 2, 5, 3, 3, 4, 4, 3, 2, 2, 1, 2, 4, 2, 4, 1, 3, 5, 3, 1, 7, 2, 3, 7, 3, 3, 1, 3, 3, 4, 7, 6, 4, 6, 5, 3, 3, 6, 9, 5, 3, 4, 3, 3, 3, 4, 1, 3, 5, 2, 5, 1, 4, 3, 5, 6, 4, 2, 3, 6, 5, 1, 3, 3, 0, 3, 4, 2, 5, 4, 4, 5, 4, 4, 8, 1, 7, 3, 2, 4, 3, 2, 5, 2, 4, 2, 4, 4, 2, 5, 6, 1, 5, 3, 5, 0, 7, 3, 2, 5, 5, 7, 3, 4, 3, 3, 4, 4, 3, 5, 4, 2, 3, 5, 6, 6, 3, 2, 4, 4, 5, 6, 2, 7, 4, 1, 5, 4, 2, 3, 5, 4, 2, 4, 2, 4, 4, 1, 2, 2, 3, 1, 2, 5, 4, 6, 5, 6, 7, 5, 7, 3, 3, 3, 2, 5, 0, 4, 3, 7, 2, 2, 5, 4, 1, 4, 2, 5, 4, 2, 6, 2, 1, 2, 4, 6, 4, 6, 4, 4, 3, 3, 4, 0, 8, 5, 3, 2, 4, 1, 6, 4, 5, 4, 2, 7, 1, 4, 3, 5, 3, 1, 6, 5, 4, 6, 4, 7, 6, 6, 10, 5, 1, 0, 5, 4, 4, 1, 1, 2, 3, 1, 3, 4, 5, 6, 5, 2, 3, 6, 3, 2, 4, 6, 2, 2, 5, 3, 2, 3, 5, 3, 5, 12, 0, 1, 10, 2, 4, 6, 1, 3, 5, 9, 2, 1, 5, 3, 0, 3, 5, 4, 3, 6, 2, 2, 6, 2, 4, 8, 2, 5, 2, 3, 5, 4, 7, 7, 4, 6, 2, 2, 7, 4, 4, 4, 5, 4, 2, 2, 4, 3, 4, 2, 3, 4, 4, 3, 3, 5, 5, 3, 3, 4, 3, 2, 4, 3, 2, 5, 4, 5, 2, 5, 0, 1, 0, 11, 2, 4, 6, 3, 6, 2, 4, 1, 2, 6, 5, 8, 2, 4, 6, 9, 4, 2, 4, 2, 2, 4, 5, 5, 1, 4, 5, 4, 3, 5, 2, 8, 5, 5, 3, 2, 6, 7, 1, 3, 4, 8, 6, 4, 12, 1, 4, 7, 4, 4, 1, 4, 5, 3, 4, 4, 3, 4, 4, 2, 5, 5, 4, 4, 6, 4, 4, 2, 3, 4, 2, 3, 5, 3, 1, 2, 2, 5, 1, 5, 4, 0, 2, 2, 2, 3, 2, 4, 5, 5, 8, 5, 4, 2, 6, 2, 4, 2, 5, 3, 4, 7, 5, 3, 4, 3, 3, 1, 3, 5, 3, 4, 3, 3, 6, 8, 5, 5, 6, 5, 3, 6, 5, 4, 7, 6, 6, 5, 4, 5, 4, 5, 3, 5, 3, 2, 5, 3, 3, 2, 2, 4, 3, 6, 6, 4, 4, 5, 7, 4, 1, 3, 6, 6, 6, 5, 4, 2, 8, 3, 3, 4, 1, 2, 3, 3, 6, 3, 4, 3, 6, 7, 3, 3, 2, 3, 3, 6, 10, 5, 4, 2, 8, 1, 5, 4, 10, 4, 1, 3, 2, 1, 7, 5, 4, 5, 9, 1, 2, 3, 3, 4, 3, 4, 8, 5, 3, 4, 5, 4, 6, 3, 7, 6, 5, 2, 3, 1, 7, 8, 1, 3, 5, 2, 3, 6, 3, 2, 2, 5, 3, 3, 4, 6, 4, 8, 7, 2, 6, 3, 4, 4, 3, 5, 4, 1, 8, 5, 6, 8, 4, 2, 2, 5, 6, 4, 5, 3, 3, 5, 3, 3, 4, 5, 5, 3, 5, 3, 1, 2, 4, 6, 1, 2, 5, 3, 3, 2, 2, 4, 4, 7, 7, 0, 3, 4, 5, 3, 4, 4, 5, 6, 6, 7, 2, 3, 6, 3, 4, 6, 4, 3, 2, 2, 3, 3, 3, 4, 2, 6, 7, 3, 5, 1, 2, 1, 2, 2, 8, 2, 1, 4, 2, 4, 4, 4, 2, 3, 4, 4, 4, 4, 4, 3, 4, 3, 2, 3, 2, 5, 2, 2, 4, 2, 3, 6, 1, 7, 2, 1, 4, 1, 4, 3, 6, 7, 2, 5, 4, 4, 4, 3, 4, 5, 6, 6

Controls · from schema()

Distribution

How many

10 5000

Sampling noise falls as 1/√n. At a hundred draws the bars wobble visibly around the curve; at five thousand they sit on it.

Rate λ

0.01 500

Poisson: the mean number of events. Exponential: the arrival rate, so the mean wait is 1/λ.

Shape α

0.05 50

Pareto tail index, Gamma shape k, Beta’s first shape, and Zipf’s exponent s. Smaller is heavier-tailed.

Shape β

0.05 50

Beta’s second shape. α = β = 1 is the uniform distribution.

Scale

0.01 100

Gamma’s θ, Cauchy’s half-width γ, and Pareto’s minimum x_m — the point its tail starts from.

Trials n

1 500

Binomial only. Capped at 500 because the sampler inverts the CDF, which costs about n·p steps a draw.

Probability p

0.001 0.999

Binomial only: the chance each try succeeds.

Ranks N

2 2000

Zipf only: how many ranks there are. Word frequency lists behave like this with s ≈ 1.

Histogram bars

4 60

Decimal places

0 6

Entropy source

Picking a source changes the story on the receipt, never the quality: anything below class A is mixed with the OS CSPRNG before it reaches you. This result actually came from NIST Randomness Beacon.

seed token NE4QKMQ8890KQSXNJH03KQHB4C

Re-render keeps this token. Generate replaces it. That is the whole idea.

entropy in

16 bytes

stream used

34,237 bytes

rejections

duration

6,639 µs

distribution

Poisson — arrivals in a fixed window

parameters

λ = 4

histogram

{"discrete":true,"lo":0,"hi":13,"width":1,"counts":[15,68,171,199,208,154,86,57,26,6,7,1,2],"expected":[18.316,73.263,146.525,195.367,195.367,156.293,104.196,59.54,29.77,13.231,5.292,1.925,0.642],"outside":0}

observed mean

3.89

observed variance

3.73

expected mean

expected variance

mean at tenth

3.85

method

Knuth: multiply uniforms until the product falls below e^(−λ). Exact, and costs about λ + 1 draws.

Seed receipt

Class B · Public beacon

NIST Beacon pulse #2,005,732 — chained to every pulse before it, and signed.

Published openly — anyone can look this value up after the fact.

source: NIST Randomness Beacon
observed: 2026-10-30T11:14:00+00:00
latency: 77ms
mixed with: csprng — always

https://beacon.nist.gov/beacon/2.0/chain/2/pulse/2005732 ↗

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The token is the seed, so this link recomputes the result rather than looking it up. No database, and it works for as long as v1 does.

This page and /api/v1/g/numbers.distribution run the same code. There is no private endpoint.

Distribution Lab

Result

Measurements

Seed receipt

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