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How to use
The Poisson distribution models a count of events in a fixed time or space interval when the expected mean λ is known and a constant-rate, independent-event model is appropriate.
- Enter mean λ from 0 to 1000.
- Enter integer event count k from 0 to 10,000.
- Calculate P(X=k), P(X≤k), P(X≥k), mean, variance and standard deviation.
How it is calculated
Example
Example: with λ=4 and k=2, P(X=2)≈14.65% and P(X≤2)≈23.81%.
Important notes
The calculator uses numerically stable cumulative-probability formulas instead of naively summing a large number of terms. Whether the result is a good real-world model depends on the Poisson assumptions fitting the data.
Worked examples and interpreting results
A mean of two events per fixed interval under a Poisson model gives probability of no events about 13.534%. Both mean and variance equal 2 in this model.
| Case | Calculation | Result |
|---|---|---|
| Probability of zero events | e⁻² × 100 | 13.533528 % |
| Variance | Var(X) = λ | 2 |
How to check the result
Enter mean 2 and count k=0. Assume constant rate and independent events over the interval; changing rates or clustering may make Poisson inappropriate.
Frequently asked questions
Exact versus at least?
P(X=k) is one count, while P(X≥k) includes k and every larger count.
What happens at λ=0?
Then X=0 with probability 1 and every positive count has probability 0.
When can Poisson approximate a binomial distribution?
The approximation is commonly used for many trials and a small success probability while λ=np remains moderate. Accuracy depends on the actual values, so it does not always replace an exact binomial calculation.