Calculator Inputs
Responsive 3 / 2 / 1 field layoutExample Data Table
Example scenario: a help desk averages 2.5 tickets per hour. The table shows probabilities for selected ticket counts in one hour.
| x | P(X = x) | P(X ≤ x) | P(X ≥ x) |
|---|---|---|---|
| 0 | 0.082085 | 0.082085 | 1.000000 |
| 1 | 0.205212 | 0.287297 | 0.917915 |
| 2 | 0.256516 | 0.543813 | 0.712703 |
| 3 | 0.213763 | 0.757576 | 0.456187 |
| 4 | 0.133602 | 0.891178 | 0.242424 |
| 5 | 0.066801 | 0.957979 | 0.108822 |
Formula Used
Probability mass function: P(X = x) = e-λ × λx / x!
Cumulative probability: P(X ≤ x) = Σ P(X = k) for all integers from 0 through x.
Expected count: λ = rate × interval when the average arrival rate is provided instead of the direct expected count.
Distribution properties: for a Poisson model, the mean and variance are both equal to λ, and the standard deviation is √λ.
How to Use This Calculator
- Select direct λ if you already know the expected number of events for the chosen period.
- Select rate × interval when you know the average rate and want the calculator to derive λ.
- Enter the target count x to evaluate the exact probability of observing that count.
- Choose the table start and end values to generate a probability table for nearby outcomes.
- Set the display precision, submit the form, then review the summary cards, table, and export options.
Frequently Asked Questions
1. What does the Poisson PMF measure?
It measures the probability of seeing exactly x events in a fixed interval when events occur independently and the average rate stays constant.
2. When should I use a Poisson model?
Use it for count data such as calls, defects, arrivals, or failures when events are rare, independent, and tracked over equal windows.
3. Why are mean and variance the same?
A standard Poisson distribution is defined so both its expected value and variance equal λ. That equality is a core identifying property.
4. What is the difference between PMF and CDF?
The PMF gives the probability of exactly one count. The CDF gives the probability of that count or anything smaller.
5. Can λ be zero?
Yes. If λ equals zero, the only possible outcome is x = 0 with probability one, and all other counts have probability zero.
6. What does P(X ≥ x) mean here?
It is the upper-tail probability of observing x or more events. This is useful when you want to test unusually high counts.
7. Why does the table need integer x values?
Poisson distributions model counts. Counts are discrete whole numbers, so fractional values such as 2.7 events are not valid outcomes.
8. What can I export from this page?
After calculating, you can export a CSV summary with the table values or create a PDF report containing the main metrics and distribution table.