Calculator Inputs
Formula Used
The Poisson model describes counts when events occur independently at a constant average rate. The expected count is λ.
Upper inclusive tail: P(X ≥ k) = Σ P(X = i), for i = k to ∞
Upper strict tail: P(X > k) = P(X ≥ k + 1)
Lower inclusive tail: P(X ≤ k) = Σ P(X = i), for i = 0 to k
Lower strict tail: P(X < k) = P(X ≤ k - 1)
When you choose rate × exposure mode, the calculator first computes λ = rate × exposure. It then evaluates the requested exact tail probability by summing the relevant Poisson probabilities.
How to Use This Calculator
- Select either direct λ input or rate × exposure input.
- Enter your expected count, or enter rate and exposure.
- Set the threshold count k you want to test.
- Choose the tail rule: at least, greater than, at most, or less than.
- Set α if you want a one-sided significance interpretation.
- Click the calculate button to display the result above the form.
- Use the CSV or PDF buttons to export the summary.
Example Data Table
| Scenario | λ | Threshold | Tail rule | Probability |
|---|---|---|---|---|
| Website outages per month | 2.4 | 5 | P(X ≥ 5) | 0.095869 |
| Support escalations per day | 7.0 | 10 | P(X ≥ 10) | 0.169504 |
| Defects per batch | 12.5 | 8 | P(X ≤ 8) | 0.124916 |
| Claims per hour | 18.0 | 20 | P(X > 20) | 0.269280 |
Interpretation Notes
A very small upper-tail probability suggests the observed count is unusually high under the assumed mean. A very small lower-tail probability suggests the observed count is unusually low. These probabilities are one-sided and depend completely on the quality of your chosen λ.
FAQs
1. What does a Poisson tail probability measure?
It measures the chance that a Poisson count falls in a selected tail of the distribution, such as at least, greater than, at most, or less than a threshold.
2. When should I use a Poisson model?
Use it for independent event counts over fixed exposure when the average rate is stable. Typical examples include arrivals, failures, defects, and incidents.
3. What is the difference between P(X ≥ k) and P(X > k)?
P(X ≥ k) includes the threshold value k itself. P(X > k) excludes it and starts at the next count, k + 1.
4. Why are the mean and variance both equal to λ?
That equality is a defining property of the Poisson distribution. It makes λ both the expected count and the variance of the count process.
5. How is rate × exposure mode useful?
It helps when you know an average rate per unit and a time, distance, or batch window. The calculator multiplies them to get λ.
6. Is the normal approximation always reliable?
No. It improves as λ grows, especially above 10. For smaller expected counts or extreme tails, the exact Poisson result is usually safer.
7. Does a small tail probability prove causation?
No. It only says the observed count is unusual under the assumed Poisson model and λ. Model misspecification can create misleading tail probabilities.
8. Can I use this as a one-sided p-value calculator?
Yes. If your null model is Poisson with mean λ, the selected tail probability can be read as a one-sided p-value.