Poisson Distribution Calculator
Compute exact or approximate probabilities for count data where events occur independently at an average rate λ.
What Is a Poisson Distribution?
The Poisson distribution models the number of times an event occurs in a fixed interval of time, area, distance, or volume when those events happen independently and at a roughly constant average rate. It is a core distribution in statistics, probability, data science, reliability engineering, operations research, and quality control.
Typical examples include customer arrivals per minute, typing errors per page, system failures per month, or calls to a help desk per hour.
Poisson Formula and Key Terms
If X follows a Poisson distribution with mean rate λ, then the probability of seeing exactly k events is:
P(X = k) = e-λ λk / k!, for k = 0, 1, 2, ...
- λ: expected number of events in the interval.
- k: observed event count.
- P(X ≤ k): cumulative probability up to k.
- P(X ≥ k): upper-tail probability (k or more events).
A useful property: for a Poisson random variable, the mean and variance are both equal to λ.
How to Use This Calculator
1) Set the average rate (λ)
Enter your expected event frequency in the selected interval. If you expect 6 arrivals per hour on average, set λ = 6.
2) Choose the probability type
- P(X = k) for an exact count.
- P(X ≤ k) for cumulative probability up to k.
- P(X ≥ k) for at least k events.
- P(a ≤ X ≤ b) for a bounded range.
3) Enter count values
Counts must be non-negative integers. If you select a range, provide both lower and upper bounds.
When the Poisson Model Is Appropriate
- Events are independent.
- The average rate is fairly constant in the interval.
- Two events are unlikely to happen at exactly the same instant.
- You are counting occurrences, not measuring continuous values.
If your data are overdispersed (variance much larger than mean), consider alternatives like negative binomial models.
Worked Example
Suppose a website receives an average of 4 support tickets per hour (λ = 4).
- Probability of exactly 2 tickets in an hour: P(X = 2).
- Probability of at most 2 tickets: P(X ≤ 2).
- Probability of 6 or more tickets: P(X ≥ 6).
This calculator gives all three with one interface, making scenario analysis quick and consistent.
Common Mistakes to Avoid
- Using negative or decimal values for event counts.
- Confusing the interval for λ (per minute vs per hour).
- Applying Poisson to non-independent events.
- Interpreting tiny probabilities as impossible events.
Final Notes
This Poisson probability tool is useful for queueing analysis, reliability studies, capacity planning, and hypothesis checks involving rare events. For very large rates, the calculator uses a normal approximation for cumulative values to keep computation fast and stable.