The Poisson distribution calculator computes the probability of observing k events in a fixed interval when the average rate is λ. It applies to rare, independent events: radioactive decay, customer arrivals, website errors, and goals in a soccer match.
Poisson Distribution
Events per interval
Step-by-Step
Formula Reference
Mean = Variance = λ Standard deviation = √λ
How to Use the Poisson Distribution Calculator
The Poisson distribution is the go-to model for counting rare, independent events in a fixed interval when you know the average rate. Unlike binomial, there's no fixed upper limit on k.
Step 1: Enter λ and k
Enter the average rate λ (average number of events per interval) and k (the specific number of events you want to find the probability for). Or select an application example to pre-fill λ.
Example: Call Center
A call center receives 2.5 calls per minute on average (λ=2.5). What's the probability of exactly k=4 calls in one minute? P(X=4) = e^(−2.5) × 2.5⁴ / 4! = 0.0821 × 39.0625 / 24 = 0.1336, about 13.4%.
Cumulative Probabilities
P(X≤k) is the probability of at most k events. For emergency planning, P(X≥k) — probability of at least k events — is often what you need. Example: P(X≥5 calls in a minute) = 1 − P(X≤4) tells you how often you're overwhelmed.
FAQ
What is the Poisson distribution?
The Poisson distribution models the number of events occurring in a fixed interval when events occur independently at a constant average rate λ. It applies to rare events: customer arrivals, radioactive decay, website errors, and goals scored in a soccer match.
What is the Poisson probability formula?
P(X=k) = (e^(-λ) × λ^k) / k!, where λ is the average rate, k is the number of events, and e ≈ 2.718. Both the mean and variance of a Poisson distribution equal λ.
When can I use a Poisson distribution?
Use Poisson when: events are independent, the average rate λ is constant, two events can't occur at exactly the same instant, and you're counting discrete events in a continuous interval of time, space, or volume.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
How does Poisson relate to binomial?
The Poisson distribution is a limiting case of the binomial when n is large and p is small, with λ = np. If n > 50 and p < 0.1, the Poisson approximation to the binomial is very accurate.