Description
Compare matched-size Cauchy and Pareto Type I samples with visible derived sampler seeds, observed extremes, medians, and linearly interpolated empirical quantiles; preserve Pareto sampler 10-significant-digit rounding and epsilon tail clipping, and do not report a finite Cauchy population mean or variance.
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.
Compare matched-size Cauchy and Pareto Type I samples with visible derived sampler seeds, observed extremes, medians, and linearly interpolated empirical quantiles; preserve Pareto sampler 10-significant-digit rounding and epsilon tail clipping, and do not report a finite Cauchy population mean or variance.
When to use this tool
Use this generator to create reproducible heavy tail comparison 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 cauchyLocation, cauchyScale, paretoScale, paretoShape, count, percentiles 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, cauchy, pareto. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.
Derive separate seeds and sample Cauchy and Pareto distributions, then return extrema, scaled median calculation, and requested percentile summaries for each batch.1
Delegated operations used for this result: cauchy sampler; pareto sampler. Their calculation or parsing conventions apply to the corresponding result fields.2, 3
| Input field | Meaning and units | Accepted input |
|---|---|---|
| cauchyLocation | Cauchy Location | Required. Finite number. |
| cauchyScale | Cauchy Scale | Required. Finite number; strictly above 0. |
| paretoScale | Pareto Scale | Required. Finite number; strictly above 0. |
| paretoShape | Pareto Shape | 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 2048. |
| percentiles | Percentiles | Required. Array of Finite number; minimum 0; maximum 100.; minimum items 1; maximum items 16. |
| 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.cauchyLocation | Cauchy Location | Required. Finite number. |
| model.cauchyScale | Cauchy Scale | Required. Finite number; strictly above 0. |
| model.paretoScale | Pareto Scale | Required. Finite number; strictly above 0. |
| model.paretoShape | Pareto Shape | Required. Finite number; strictly above 0. |
| cauchy | Cauchy | Required. Object; no unknown keys. |
| cauchy.samplerSeed | Sampler Seed | Required. Text; minimum UTF-16 length 1. |
| cauchy.samples | Samples | Required. Array of Finite number.; minimum items 1; maximum items 2048. |
| cauchy.minimum | Minimum | Required. Finite number. |
| cauchy.maximum | Maximum | Required. Finite number. |
| cauchy.median | Sample median in the same units as values. | Required. Finite number. |
| cauchy.quantiles | Quantiles | Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16. |
| cauchy.quantiles[].percentile | Percentile | Required. Finite number; minimum 0; maximum 100. |
| cauchy.quantiles[].value | Value | Required. Finite number. |
| pareto | Pareto | Required. Object; no unknown keys. |
| pareto.samplerSeed | Sampler Seed | Required. Text; minimum UTF-16 length 1. |
| pareto.samples | Samples | Required. Array of Finite number.; minimum items 1; maximum items 2048. |
| pareto.minimum | Minimum | Required. Finite number. |
| pareto.maximum | Maximum | Required. Finite number. |
| pareto.median | Sample median in the same units as values. | Required. Finite number. |
| pareto.quantiles | Quantiles | Required. Array of Object; no unknown keys.; minimum items 1; maximum items 16. |
| pareto.quantiles[].percentile | Percentile | Required. Finite number; minimum 0; maximum 100. |
| pareto.quantiles[].value | Value | Required. Finite number. |
| 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.
{
"input": {
"count": 4,
"cauchyLocation": 0,
"cauchyScale": 1,
"paretoScale": 1,
"paretoShape": 2,
"percentiles": [
25,
75
],
"seed": 42
},
"output": {
"model": {
"cauchyLocation": 0,
"cauchyScale": 1,
"paretoScale": 1,
"paretoShape": 2
},
"cauchy": {
"samplerSeed": "cauchy:2581720956",
"samples": [
-0.9827371429979488,
-2.4105147195888246,
-6.616965306583044,
68.51463976143775
],
"minimum": -6.616965306583044,
"maximum": 68.51463976143775,
"median": -1.6966259312933867,
"quantiles": [
{
"percentile": 25,
"value": -3.4621273663373793
},
{
"percentile": 75,
"value": 16.391607083110976
}
]
},
"pareto": {
"samplerSeed": "pareto:1925393290",
"samples": [
3.267891477,
1.039048033,
1.210292686,
1.200343332
],
"minimum": 1.039048033,
"maximum": 3.267891477,
"median": 1.205318009,
"quantiles": [
{
"percentile": 25,
"value": 1.1600195072500001
},
{
"percentile": 75,
"value": 1.7246923837500001
}
]
},
"randomness": {
"seed": 42,
"source": "seed"
}
}
}Limitations and assumptions
- Heavy-tail samples can contain extreme finite values and numerical generation can fail. Finite-sample quantiles are descriptive, not population moment estimates or proof that real data follows either distribution.
- 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
Similar or alternative tools
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- Random Overdispersed Count Experiment Generator
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.