Poisson Distribution
PMF and CDF for rare-event counts
About this calculator
The Poisson Distribution Calculator gives the probability of a count of rare, independent events given an average rate λ. It returns the exact probability P(X = k), the cumulative tails P(X ≤ k) and P(X ≥ k), plus the mean, variance, standard deviation, and most likely count — and draws the probability bars with your selection highlighted. Useful for arrivals, defects, calls, decay counts, and other rare-event modeling.
How to use the Poisson Distribution calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- λ = 3 arrivals/hour: P(X = 2) = 0.2240
- Cumulative: P(X ≤ 2) = 0.4232 when λ = 3
- Right tail: P(X ≥ 5) = 0.1847 when λ = 3
- No events: P(X = 0) = e^−λ = 0.0498 when λ = 3
- Mean λ = 10 ⇒ variance 10, σ = 3.1623, most likely count 10
Frequently asked questions
How does the Poisson Distribution calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Poisson Distribution calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.