Description
Estimate a regular finite real expression integral with uniform samples and an empirical standard error, retaining samples and seed metadata without claiming a certified bound or improper-integral support.
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.
Estimate a regular finite real expression integral with uniform samples and an empirical standard error, retaining samples and seed metadata without claiming a certified bound or improper-integral support.
When to use this tool
Use this generator to create reproducible monte carlo integration 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 expression, variable, lowerBound, upperBound, 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 abscissas, values, estimate, standardError, sampleCount. Use the conventions below and keep answer-bearing fields separate when presenting an exercise.
Parse the bounded mathematical expression, verify finite endpoint evaluations, and sample abscissas across the interval. Accumulate the mean and sample variance online, returning signed span times mean and absolute span times the standard error of the mean.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| expression | Finite real expression supported by the existing scientific parser; explicit multiplication, trigonometry in radians, log base 10, ln natural log. | Required. Text; minimum UTF-16 length 1; maximum UTF-16 length 1024. |
| variable | Variable | Optional. Text; minimum UTF-16 length 1; maximum UTF-16 length 64; default "x". |
| lowerBound | Lower Bound | Required. Finite number. |
| upperBound | Upper Bound | Required. Finite number. |
| count | Requested count; the mechanics explain whether one draw expands into several returned records. | Required. Safe integer; minimum 2; maximum 4096. |
| 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 |
|---|---|---|
| abscissas | Abscissas | Required. Array of Finite number.; minimum items 2; maximum items 4096. |
| values | Values | Required. Array of Finite number.; minimum items 2; maximum items 4096. |
| estimate | Estimate | Required. Finite number. |
| standardError | Empirical standard error under the independent uniform-sampling finite-variance model; not a certified error bound. | Required. Finite number; minimum 0. |
| sampleCount | Sample Count | Required. Safe integer; minimum 2; maximum 4096. |
| 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": {
"expression": "x^2",
"lowerBound": 0,
"upperBound": 1,
"count": 6,
"seed": 42
},
"output": {
"abscissas": [
0.6011037519201636,
0.44829055899754167,
0.8524657934904099,
0.6697340414393693,
0.17481389874592423,
0.5265925421845168
],
"values": [
0.3613257205724976,
0.20096442528632838,
0.7266979290712341,
0.44854368626271085,
0.030559899194750252,
0.27729970548435207
],
"estimate": 0.34089856097864557,
"standardError": 0.09674405339469955,
"sampleCount": 6,
"randomness": {
"seed": 42,
"source": "seed"
}
}
}Limitations and assumptions
- This is a finite-expression estimator, not an improper-integral solver. Reversed bounds reverse the estimate and equal bounds give zero when evaluations are valid. The reported standard error is an estimate rather than a guaranteed error bound.
- 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
- Monte Carlo Pi Estimator
Draw bounded uniform unit-square points, classify the inclusive quarter-circle boundary through the distance tool, and estimate pi as four times the observed fraction, without an accuracy guarantee.
- One-Dimensional Monte Carlo Random Walk Simulator
Simulate exact-count one-dimensional unit-step paths under an explicit forward-step probability, reusing cumulative sums and population endpoint summaries without predicting a real process.