Live model
Run a reproducible event-count experiment and compare it with exact theory
This is a Monte Carlo laboratory, not another PMF page. It generates a finite sample from a seeded Poisson process, measures sample moments and trigger frequency, then shows the amount of sampling noise against theoretical benchmarks.
| Simulated count | Frequency | Relative frequency | Threshold side |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
Draw X by multiplying seeded U(0,1) values until the product <= e^(-lambda); sample variance = sum(xi-xbar)^2/(n-1)
The simulation uses deterministic pseudo-random uniforms and the Poisson product stopping rule. It reports sample variance with the n-1 denominator, theoretical mean standard error sqrt(lambda/n), and the exact analytical threshold tail for an apples-to-apples check.
Use the simulator
Five steps for a reviewable Monte Carlo run
- State lambda for one trial. Every draw represents the same exposure window.
- Choose enough trials for the decision. More trials reduce Monte Carlo noise but increase work; they do not validate assumptions.
- Record the seed before running. The seed is part of the evidence, not a decorative input.
- Set an inclusive threshold. The run counts every simulated value at or above that count.
- Compare errors with their scale. Review mean error against sqrt(lambda/n) and alert-frequency error against the binomial Monte Carlo SE.
Five foundations
How to read a finite Poisson experiment
1. Pseudo-random is replayable
The generator is deterministic once seeded. Reproducibility supports debugging and review without making draws non-random for simulation purposes.
2. Sampling error is expected
A correct run normally misses lambda and exact tail probability. The important question is whether the miss is plausible for n.
3. Theory remains exact
The analytical Poisson tail is available without simulation. Monte Carlo is valuable for teaching, workflow tests, and models that later become more complex.
4. Seeds are not scenarios
Changing a seed samples another possible run under the same assumptions; changing lambda defines a different process scenario.
5. Moment agreement is incomplete
A sample can have a reasonable mean yet an unusual tail. That is why the page checks mean, variance, zero frequency, and one decision threshold.
Calculation anatomy
Random inputs, estimators, and default substitution
| Symbol | Meaning | Unit / rule |
|---|---|---|
| lambda | Theoretical mean per trial | events, 0-100 |
| n | Number of simulated trials | whole trials, 100-100,000 |
| s | Seed for pseudo-random uniforms | integer |
| xbar | Simulated sample mean | events per trial |
| s^2 | Sample variance | events^2, n-1 denominator |
| m | Inclusive alert threshold | whole events |
With defaults, 5,000 replayable draws are generated from lambda=3.5 using seed 24,017. The empirical alert fraction is count(Xi>=7)/5000; the benchmark is independently calculated as 1-F(6;3.5).
Deep analysis
Three ways to diagnose a simulation result
Mean error in standard-error units
Raw mean error is hard to judge. Divide it by sqrt(lambda/n); an error near one SE is ordinary, while repeated large errors suggest implementation trouble.
Variance and zero frequency
Poisson theory predicts variance lambda and P(X=0)=e^-lambda. These catch defects that a mean-only check can miss, including a generator with compressed spread.
Threshold-specific validation
An operational trigger usually lives in a tail. Compare the empirical trigger fraction with the exact tail and its binomial SE, rather than expecting decimal equality.
Decision cases
Routine and stress examples
Nightly incident capacity
A reliability team models 3.5 incidents per night and a seven-incident escalation. The seeded run tests a dashboard pipeline; the exact tail remains the service-level probability entered in policy.
Zero-rate boundary
At lambda=0 every draw must be zero, sample variance must be zero, and any positive alert threshold must have zero frequency. This boundary is a direct implementation check for stale random values.
Terms
Simulation vocabulary
- Seed
- Integer that initializes a reproducible pseudo-random sequence.
- Trial
- One independent simulated exposure window.
- Monte Carlo error
- Random difference between finite-sample output and the exact model quantity.
- Sample variance
- Spread estimator using squared deviations and an n-1 denominator.
- Empirical frequency
- Observed proportion of simulated draws meeting an outcome rule.
- Theoretical benchmark
- Analytical Poisson quantity under the same lambda and threshold.
FAQ
Questions specific to seeded event simulation
Why does the simulation need a seed?
It lets another reviewer recreate the same pseudo-random sample exactly.
Do more trials make the real model correct?
No. They reduce simulation error around the assumed model only.
Why use n-1 for variance?
The page displays the conventional unbiased finite-sample variance estimator.
Should the sample mean equal lambda?
No. Its expected deviation scale is sqrt(lambda/n).
Why compare the threshold with an exact tail?
It validates the operational quantity directly and exposes finite-sample deviation.
Can I quote the simulation frequency as exact?
No. Quote the analytical probability when available and retain simulation uncertainty separately.
Limits and evidence
What the run cannot establish
- The pseudo-random sequence is intended for analysis and teaching, not cryptography.
- Trial independence and a fixed lambda are assumed rather than inferred from observed process data.
- The 100,000-trial ceiling protects browser responsiveness; it is not a statistical recommendation.
- Only one user-selected tail is reconciled; inspect other features if they matter to the decision.
- A different seed legitimately changes empirical results while leaving theory unchanged.
Sources and related tools