Poisson Distribution Calculator
Probability of k events in an interval.
Enter your values
Enter the average rate λ you are working with.
Enter the number of events k you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- P(X = k)
- 0.104196
- P(X ≤ k)
- 0.889326
- P(X > k)
- 0.110674
- Mean and variance
- 4
- Standard deviation
- 2
Quick answer
The Poisson distribution models rare independent events at a constant average rate λ over a fixed interval.
P(X=k) = λ^k e^(−λ) / k!
At a glance
| What it does | Probability of k events in an interval. |
|---|---|
| Category | Statistics |
| Inputs needed | Average rate λ, Number of events k |
| Main output | P(X = k) |
| Formula | P(X=k) = λ^k e^(−λ) / k! |
| Cost | Free — no sign-up, no download |
How to use the Poisson Distribution Calculator
- 1Enter the average number of events per interval.
- 2Enter the event count you are testing.
- 3Read exact and cumulative probabilities.
Inputs explained
- Average rate λ
- Enter the average rate λ you are working with.
- Number of events k
- Enter the number of events k you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Average rate λ4
- Number of events k6
Results
- P(X = k)0.104196
- P(X ≤ k)0.889326
- P(X > k)0.110674
- Mean and variance4
- Standard deviation2
Frequently asked questions
When is Poisson appropriate?
Independent rare events at a steady rate: calls per hour, defects per metre.
How does it relate to binomial?
Poisson is the limit of binomial when n is large and p is small.
What do I need to enter into the Poisson Distribution Calculator?
Just 2 values: average rate λ, number of events k. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Poisson Distribution Calculator work out the answer?
The Poisson distribution models rare independent events at a constant average rate λ over a fixed interval. It applies the formula P(X=k) = λ^k e^(−λ) / k! and shows the working so you can check each step by hand.
Is the Poisson Distribution Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.