Measure event intensity with flexible lambda estimation tools. Switch between counts, probabilities, and time windows. Review distributions, confidence checks, and exportable summaries easily today.
This sample shows incident counts recorded across ten equal intervals. The sample mean equals 1.10, so the estimated lambda is 1.10 events per interval.
| Interval | Observed Count | Cumulative Events | Running Mean |
|---|---|---|---|
| 1 | 0 | 0 | 0.00 |
| 2 | 1 | 1 | 0.50 |
| 3 | 2 | 3 | 1.00 |
| 4 | 1 | 4 | 1.00 |
| 5 | 3 | 7 | 1.40 |
| 6 | 0 | 7 | 1.17 |
| 7 | 2 | 9 | 1.29 |
| 8 | 1 | 10 | 1.25 |
| 9 | 1 | 11 | 1.22 |
| 10 | 0 | 11 | 1.10 |
λ̂ = (Σxᵢ) / n
Use this when you have count observations over equal intervals.
λ̂ = k / T
Here, k is the total event count and T is total observed exposure.
P(X = 0) = e-λt, so λ = -ln(P(X = 0)) / t
This method is useful when no-event frequency is known directly.
P(X ≥ 1) = 1 - e-λt, so λ = -ln(1 - P(X ≥ 1)) / t
Use this when a first-event occurrence probability is available.
λ = 1 / E(W)
For a Poisson process, average waiting time and event rate are reciprocal.
P(X = k) = e-λt(λt)k / k!
The calculator solves this equation numerically because closed-form lambda values are not always available.
μ = λτ, Var(X) = μ, SD = √μ
For a future interval τ, the Poisson mean and variance both equal μ.
Lambda is the average number of events expected in one interval. It also controls the spread of the distribution because the variance equals the mean for a true Poisson process.
Use it when you have repeated event counts from equal time windows, pages, batches, sensors, or regions. The sample mean is the maximum likelihood estimator for Poisson lambda.
For some fixed counts, the Poisson probability rises to a peak and then falls. A single probability below that peak can intersect the curve at two separate lambda values.
A ratio near one supports Poisson-like variation. A much larger value suggests overdispersion, while a much smaller value suggests underdispersion, data smoothing, or structural constraints.
No. Poisson observations represent counts of events, so the interval data should be whole numbers. Probabilities, exposures, and waiting times may be entered as decimals.
The page uses a normal approximation for quick interpretation. It is useful for moderate samples, but exact Poisson intervals are preferred in strict statistical reporting.
It rescales lambda to your chosen projection interval. The distribution table, zero-event probability, and at-least-one probability are all calculated from this projected mean.
The exports include the current result summary and the displayed distribution table. Change the method, interval, decimals, or k limit first, then recalculate before downloading.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.