Calculator Inputs
Example Data Table
| λ | Query | Approximate Result |
|---|---|---|
| 2.40 | P(X ≤ 3) | 0.778723 |
| 3.80 | P(X = 2) | 0.161517 |
| 4.50 | P(X ≥ 6) | 0.297070 |
| 5.20 | P(3 ≤ X ≤ 6) | 0.623607 |
Formula Used
Poisson probability mass function:
P(X = k) = e-λ × λk / k!
Lower cumulative probability:
P(X ≤ k) = Σ[e-λ × λi / i!] for i = 0 to k.
Upper cumulative probability:
P(X ≥ k) = 1 - P(X ≤ k - 1) and P(X > k) = 1 - P(X ≤ k).
Interval probability:
Inclusive and exclusive ranges are built from cumulative probabilities by subtraction. For example, P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a - 1).
Distribution properties:
For a Poisson model, the mean equals λ, the variance also equals λ, and the standard deviation equals √λ.
How to Use This Calculator
- Enter the average event rate λ for one fixed observation period.
- Select the probability statement you want to evaluate.
- Provide a threshold count k, or enter interval bounds a and b.
- Choose how many decimal places you want in the output.
- Press the calculate button to show the result above the form.
- Review the summary table, distribution snapshot, and complementary probability.
- Download the displayed result as CSV or PDF when needed.
Frequently Asked Questions
1. What does λ mean in a Poisson model?
λ is the expected average number of events in one fixed interval. It must match the same time, area, distance, or volume unit as your count data.
2. When should I use a Poisson distribution?
Use it when events are counted, rare enough per interval, and reasonably independent. Common examples include defects, arrivals, failures, clicks, and incidents over constant exposure windows.
3. What is the difference between exact and cumulative probability?
Exact probability measures one specific count, such as P(X = 4). Cumulative probability adds multiple exact outcomes together, such as P(X ≤ 4) or P(X ≥ 4).
4. Why are mean and variance both equal to λ?
That equality is a defining property of the Poisson distribution. It makes the model useful for count processes where spread tends to track the average event rate.
5. Can λ be zero?
Yes. If λ = 0, the process predicts zero events with certainty. Then P(X = 0) equals 1 and every positive count has probability 0.
6. What do upper-tail probabilities help measure?
Upper tails estimate the chance of observing unusually large counts. They are useful for overload monitoring, anomaly detection, queue risk, reliability screening, and service capacity planning.
7. Can I use non-integer values for k, a, or b?
No. Event counts in a Poisson model are whole numbers. The calculator therefore accepts only non-negative integers for thresholds and interval bounds.
8. Why might a result look very small?
Rare or extreme counts can produce tiny probabilities. That does not indicate an error. It simply means the selected outcome is unlikely under the entered event rate.