Analyze Poisson event rates with precise mean and spread. Check exact and cumulative probabilities instantly. Download results and visualize distributions for reporting and teaching.
This example uses λ = 4. It helps you compare exact and cumulative probabilities for several event counts.
| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.018316 | 0.018316 |
| 1 | 0.073263 | 0.091578 |
| 2 | 0.146525 | 0.238103 |
| 3 | 0.195367 | 0.433470 |
| 4 | 0.195367 | 0.628837 |
| 5 | 0.156293 | 0.785130 |
| 6 | 0.104196 | 0.889326 |
For a Poisson process, the expected event count across an interval equals λ.
Effective λ = rate × interval
Mean = λ
Variance = λ
Standard deviation = √λ
P(X = k) = e-λ λk / k!
P(X ≤ k) is the sum of probabilities from 0 through k.
P(a ≤ X ≤ b) equals P(X ≤ b) − P(X ≤ a − 1).
Enter the average event rate per unit and the interval length.
Add an observed count if you want an exact and cumulative probability for one value.
Enter lower and upper bounds when you want an inclusive range probability.
Choose the number of decimal places, then press Calculate.
Review the result table above the form, inspect the graph, and export the output as CSV or PDF.
This calculator is useful when you model count data for independent events over a fixed interval. Common examples include website hits, system alerts, manufacturing defects, support tickets, and arrivals per minute. In those settings, the Poisson mean equals the variance, and the standard deviation is the square root of the mean.
The form accepts an event rate and interval so you can build the effective λ value from real operational data. It also lets you test a single observed count, estimate upper-tail rarity, and measure the probability that counts stay inside a chosen range. That makes it practical for forecasting, anomaly checks, capacity planning, and classroom examples.
The graph helps you see where the mass of the distribution sits, while the export tools help you keep records or share results with others. Because the layout is simple and stacked, it also works well for embedded tools, tutorials, and internal analytics pages.
The mean is the expected number of events in the selected interval. If λ equals 5, the long-run average count is five events per interval.
For a Poisson distribution, the variance equals λ by definition. Standard deviation is the square root of variance, so it becomes √λ.
Use it for counts of independent events that happen randomly over time, space, or volume, especially when you know an average rate and want probability estimates.
It calculates the exact probability for one count, the cumulative probability up to that count, the upper tail from that count, and a z score estimate.
They help you answer range questions, such as the chance of seeing between 3 and 7 arrivals during a time window.
Yes. λ is an average rate, so fractional values are common. A value like 2.4 means the process averages 2.4 events per interval.
The graph shows the probability mass function. Each bar represents the probability of observing a specific whole-number event count for your calculated λ.
CSV is useful for analysis in spreadsheets and reports. PDF is useful for sharing a clean summary with students, teammates, or stakeholders.
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.