Poisson Distribution Calculator
Enter the average event rate λ and an event count k to evaluate Poisson probabilities and distribution summaries.
P(X = k)
0.146525
P(X ≤ k)
0.238103
P(X > k)
0.761897
Mean: 4.000
Variance: 4.000
Standard Deviation: 2.000
| k | P(X = k) |
|---|---|
| 0 | 0.018316 |
| 1 | 0.073263 |
| 2 | 0.146525 |
| 3 | 0.195367 |
| 4 | 0.195367 |
| 5 | 0.156293 |
| 6 | 0.104196 |
| 7 | 0.059540 |
| 8 | 0.029770 |
| 9 | 0.013231 |
| 10 | 0.005292 |
| 11 | 0.001925 |
| 12 | 0.000642 |
| 13 | 0.000197 |
| 14 | 0.000056 |
How to Use This Calculator
- Enter the average event rate λ (events per interval).
- Provide an event count k to evaluate exact and cumulative probabilities.
- Review mean, variance, and standard deviation — all equal to λ in a Poisson process.
- Use the probability table to inspect neighbouring event counts.
Formula
P(X = k) = e−λ λk / k!
P(X ≤ k) = Σi=0k e−λ λi / i!
Mean = λ • Variance = λ
The Poisson distribution models counts of events that occur independently with a constant average rate over a fixed interval of time or space.
Full Description
Poisson processes arise in queueing systems, natural phenomena, telecommunications, and reliability analysis. The single parameter λ simultaneously represents the expected number of events and the distribution variance. Because Poisson probabilities decrease quickly in the tails, even modest λ values yield a practical upper bound on likely event counts.
When λ is large, the Poisson distribution is well-approximated by a normal distribution with identical mean and variance, while small λ highlights the discrete and skewed nature of the process.
Frequently Asked Questions
Can k be negative or fractional?
No. k must be a non-negative integer because it represents a count of events.
What if the variance exceeds λ?
Over-dispersed data may suit a negative binomial model, which allows variance greater than the mean.
How do I approximate cumulative probabilities quickly?
For large λ, use the normal approximation with continuity correction; otherwise rely on Poisson sums as shown here.
Does the interval length matter?
λ should reflect the expected count over the interval of interest. Adjust λ if you change the observation interval.