How to use the monte carlo calculator
Prepare the values for this calculation
Estimate a one-dimensional definite integral by reproducible uniform Monte Carlo sampling. Use the formula below to check that it answers your question. Read the input units before transferring figures from a statement, measurement or dataset.
Enter and check each input
- Integrand: choose Constant c, x, x², sin(x), e^(−x). Match this selection to the data you enter.
- Constant c: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- Lower bound a: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- Upper bound b: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- Samples: enter a value at least 100 and no more than 1000000 and a whole number. Use the unit named in the label or selected above. Do not leave this required field blank.
- Deterministic seed: enter a value at least 1 and no more than 2147483646 and a whole number. Use the unit named in the label or selected above. Do not leave this required field blank.
The input table gives a complete example. Keep units and time periods consistent across related fields.
Calculate and resolve any input error
Choose Calculate after entering the values. Changes also update the answer after a short pause. If an error appears, follow its message, correct the field and calculate again. Reset restores the example values.
Interpret the answer
Check Monte Carlo integral estimate first. With the unchanged example inputs, it is 0.336254924394. Compare the additional metrics with the primary result; each has its own label and unit.
Compare and keep a useful result
For a controlled comparison, keep every other example input unchanged and change Lower bound a from 0 to 0.1. The resulting Monte Carlo integral estimate is 0.335554306095. This illustrates the equation’s response to that input, rather than a forecast of real events.
Read the chart’s axis labels and data table. A scenario curve changes one stated input while holding the others fixed; it does not add external observations. Use Download graph to save an available chart as a PNG.
Use Copy result or Download CSV to keep the answer. Share tool shares the page address without placing your input values in the URL. Recheck inputs and assumptions before using the result for a decision.
Monte Carlo Calculator formula
For U~Uniform(a,b), integral ≈(b−a)·mean[f(U)]; Monte Carlo SE=(b−a)·s[f(U)]/√n.
Inputs and units
| Input | Format / range | Example |
|---|---|---|
| Integrand | Select an option | x² |
| Constant c | Number | 2 |
| Lower bound a | Number | 0 |
| Upper bound b | Number | 1 |
| Samples | minimum 100; maximum 1000000; whole number | 10000 |
| Deterministic seed | minimum 1; maximum 2147483646; whole number | 12345 |
Worked example
Using the example inputs above, the monte carlo integral estimate is 0.336254924394.
- Monte Carlo integral estimate
- 0.336254924394
- Monte Carlo SE
- 0.002996607765
- Samples
- 10000
- Seed
- 12345
Results and assumptions
Reproducible pseudorandom uniform Monte Carlo estimate; the reported SE is simulation error, not quadrature bias.
Fix an input or unexpected result
Constant c is required. Enter constant c using the format and bounds in the input table. Zero and an empty field are different: use zero only if it is allowed and represents your data.
If your answer differs from the example, choose Reset and compare every field, including selected units and options. Check percentage versus decimal entry and whether values describe the same period. Round only the final displayed answer when reproducing the formula.
If a discrepancy remains, report the calculator issue with the page address, non-sensitive example inputs, actual result, expected result and a reference for your calculation.