Random Overdispersed Count Experiment Generator

Description

Compare equal-size binomial and beta-binomial batches at the same beta-derived mean probability, using visible derived string-seeded sampler substreams; report observed population means/variances and theoretical variances, without claiming every realized batch is more dispersed or fitting real data.

Synthetic statistical data separates a specified generating model from the values observed in a finite draw. A model parameter, a sample summary, and the result of a fitted analysis answer different questions; retaining them separately makes numerical and statistical checks easier to interpret.

When to use this tool

Use this generator to create reproducible overdispersed count experiment cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply trials, alpha, beta, count in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read model, binomial, betaBinomial. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.

Derive separate sampler seeds for ordinary binomial and beta-binomial draws with the same theoretical mean probability alpha/(alpha+beta). Return observed population moments and theoretical variances, including the beta-binomial dispersion multiplier.^[1]

Delegated operations used for this result: binomial sampler; beta binomial sampler. Their calculation or parsing conventions apply to the corresponding result fields.^{[2, 3]}

Input field	Meaning and units	Accepted input
trials	Trials	Required. Safe integer; minimum 0; maximum 1000000.
alpha	Alpha	Required. Finite number; strictly above 0.
beta	Beta	Required. Finite number; strictly above 0.
count	Requested count; the mechanics explain whether one draw expands into several returned records.	Required. Safe integer; minimum 1; maximum 512.
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
model	Model	Required. Object; no unknown keys.
model.trials	Trials	Required. Safe integer; minimum 0.
model.alpha	Alpha	Required. Finite number; strictly above 0.
model.beta	Beta	Required. Finite number; strictly above 0.
model.meanProbability	Mean Probability	Required. Finite number; minimum 0; maximum 1.
model.theoreticalMean	Theoretical Mean	Required. Finite number; minimum 0.
binomial	Binomial	Required. Object; no unknown keys.
binomial.samplerSeed	Sampler Seed	Required. Text; minimum UTF-16 length 1.
binomial.samples	Samples	Required. Array of Safe integer; minimum 0.; minimum items 1; maximum items 512.
binomial.observedMean	Observed Mean	Required. Finite number; minimum 0.
binomial.observedVariance	Observed Variance	Required. Finite number; minimum 0.
binomial.theoreticalVariance	Theoretical Variance	Required. Finite number; minimum 0.
betaBinomial	Beta Binomial	Required. Object; no unknown keys.
betaBinomial.samplerSeed	Sampler Seed	Required. Text; minimum UTF-16 length 1.
betaBinomial.samples	Samples	Required. Array of Safe integer; minimum 0.; minimum items 1; maximum items 512.
betaBinomial.observedMean	Observed Mean	Required. Finite number; minimum 0.
betaBinomial.observedVariance	Observed Variance	Required. Finite number; minimum 0.
betaBinomial.theoreticalVariance	Theoretical Variance	Required. Finite number; minimum 0.
randomness	Replay metadata: source is seed or context; seed is the normalized 32-bit seed or null for a context source.	Required. Object; no unknown keys.
randomness.seed	Optional numeric replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "trials": 6,
    "alpha": 2,
    "beta": 2,
    "count": 4,
    "seed": 42
  },
  "output": {
    "model": {
      "trials": 6,
      "alpha": 2,
      "beta": 2,
      "meanProbability": 0.5,
      "theoreticalMean": 3
    },
    "binomial": {
      "samplerSeed": "binomial:2581720956",
      "samples": [
        2,
        5,
        2,
        0
      ],
      "observedMean": 2.25,
      "observedVariance": 3.1875,
      "theoreticalVariance": 1.5
    },
    "betaBinomial": {
      "samplerSeed": "beta-binomial:1925393290",
      "samples": [
        5,
        6,
        0,
        3
      ],
      "observedMean": 3.5,
      "observedVariance": 5.25,
      "theoreticalVariance": 3
    },
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Overdispersion is a model comparison, not a guarantee that every finite sample has greater observed variance. Work is bounded by trials times count and rejected sampling parameters return a structured error.
A replay requires the same complete input and implementation. Randomized construction does not guarantee uniform coverage of every feasible output; finite samples can repeat unless a distinct-selection rule is explicitly stated.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Overdispersed Count Experiment Generator — tool.io
Calculation or validation: probability/binomial-sampler — tool.io
Calculation or validation: probability/beta-binomial-sampler — tool.io

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