Enter Distribution Inputs
Example Data Table
| Scenario | Rate λ | Exposure | Requested Output | Interpretation |
|---|---|---|---|---|
| Calls arriving each minute | 3.2 | 1 | P(X = 4) | Chance of exactly four calls in one minute |
| Defects per sheet batch | 1.8 | 2 | P(X ≤ 3) | Chance of at most three defects in two batches |
| Website signups per hour | 5.5 | 0.5 | P(1 ≤ X ≤ 4) | Chance of one to four signups in thirty minutes |
Formula Used
Mean for the chosen interval: μ = λ × exposure
Exact probability: P(X = x) = e-μ × μx / x!
Cumulative probability: P(X ≤ x) = Σ P(X = i), for i = 0 to x
Interval probability: P(a ≤ X ≤ b) = Σ P(X = i), for i = a to b
Upper tail: P(X ≥ x) = 1 − P(X ≤ x − 1)
Distribution properties: Mean = μ, Variance = μ, Standard Deviation = √μ
How to Use This Calculator
- Enter the average event rate λ for one base interval.
- Enter the exposure multiplier for a longer or shorter interval.
- Provide the exact count x for single-value probability.
- Provide the cumulative count x for the left-tail probability.
- Enter lower and upper bounds for interval probability.
- Choose the maximum x to display in the output table.
- Press the calculate button to generate the result block.
- Use the CSV or PDF buttons to export results.
Frequently Asked Questions
1. What does the Poisson distribution measure?
It models the probability of a given number of events occurring in a fixed interval when events happen independently and at a stable average rate.
2. When should I use this calculator?
Use it for rare or count-based events such as arrivals, defects, calls, failures, or signups when the average rate is known and outcomes are nonnegative integers.
3. What is the difference between λ and μ?
λ is the average rate for one base interval. μ is the adjusted mean for the selected exposure. This calculator computes μ by multiplying λ by exposure.
4. Why are mean and variance the same?
For a Poisson model, both the expected count and the variance equal μ. That property is one reason the distribution is useful for modeling random event counts.
5. What does the interval probability represent?
It gives the chance that the event count falls between the chosen lower and upper bounds, including both endpoints.
6. Can I use decimal values for event counts?
No. Event counts must be whole numbers because the Poisson distribution applies to discrete counts such as 0, 1, 2, and so on.
7. What does P(X ≥ 1) tell me?
It shows the probability that at least one event occurs in the selected interval. This is often useful in reliability, demand, and queue analyses.
8. Does this calculator support exports?
Yes. After calculation, you can export the displayed metrics and distribution table to CSV or generate a PDF-style report directly from the page.