Poisson Distribution Calculator
Enter the average event rate for an interval and the count or range you care about to find exact, cumulative or tail probabilities.
See the Poisson probability
Enter your values, or try an example.
Your results and calculation will appear here.
Show probability table
| Event count (x) | P(X = x) | P(X ≤ x) |
|---|
Show calculation
How to use this calculator
- Enter the average number of events in one interval (λ).
- Choose your probability event and enter a whole-number count or range.
- Calculate to see probability, expected count and the highlighted PMF.
Worked example: calls per hour
This example uses sample values, not your current inputs.
mode: at-most · rate: 4 · x: 6
Probability = 0.8893; Percentage = 88.9326%
The probability of 6 or fewer events is 88.9326%.
Example calculation
- P(X = k) = e⁻λ × λᵏ / k!.
- λ = 4; selected event: P(X ≤ 6).
- P(X ≤ 6) = Σ[k = 0…6] e⁻4 × 4^k / k!. The CDF includes every count from 0 through 6.
- Probability ≈ 0.8893; percentage ≈ 88.9326%.
- Mean = variance = λ = 4.
- SD = √4 = 2.
Formula & assumptions
P(X = k) = e⁻λ λᵏ / k!; mean = variance = λ; SD = √λ.
In the formula, λ is the expected number of events in the selected interval, k is the event count labeled x in the calculator, and e is Euler’s number (approximately 2.71828). The symbol k! means k factorial, with 0! = 1.
- Events occur independently at a constant average rate.
- The rate and observed count refer to the same interval.
- Counts are inclusive for at most, at least and between. Outside excludes both endpoints.
- Rate from 0 to 10,000; event counts up to 100,000. The chart may show only a central window.
Poisson Distribution and Event Rates
A Poisson distribution models the number of events in a fixed interval when events occur independently at a constant average rate. If the event rate is r per unit of time and the interval lasts t units, the expected count is λ = r × t. Use this λ value to calculate event-count probabilities.
For example, if an event occurs at an average rate of three times per hour, the expected count over two hours is λ = 3 × 2 = 6. Enter λ = 6 to calculate the probability of a particular number of events during that two-hour interval.
λ = r × t
- r — average event rate per unit of time.
- t — length of the interval in matching units.
- λ — expected number of events during the interval.
Sources
Reviewed: October 2026
Common questions
When might a Poisson model be a poor fit?
A Poisson model may be a poor fit when events influence one another, the event rate changes substantially, or observed counts are much more variable than the model predicts. For successes in a fixed number of independent yes-or-no trials with a constant success probability, a binomial distribution may be more appropriate.
What is lambda in a Poisson distribution?
Lambda is the average number of events in the interval you are studying. A rate of 4 calls per hour means λ = 4 for one hour or λ = 2 for 30 minutes, assuming the same constant rate. Use the interval that matches your event count.
Can I use this as a Poisson process calculator?
You can calculate event-count probabilities for a Poisson process using the expected number of events in a specified interval. Enter λ for that interval. Waiting times between events follow a related exponential distribution and are not calculated here.
Probability reference table
Explore the Poisson Distribution Table to compare probabilities across parameter columns.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.