Poisson Distribution Calculator
Calculate the probability of exactly k events for an average event rate λ.
Poisson probability formula
P(X=k) = e^(−λ) × λ^k / k!. The Poisson distribution models event counts over a fixed interval when events occur independently at an average rate.
How Poisson Probability Works
The Poisson distribution models the number of events occurring in a fixed interval when events happen independently at an average rate.
Poisson Example
If a help desk receives an average of 3 calls per hour, the calculator can estimate the probability of receiving exactly 5 calls in the next hour.
When to Use It
Poisson models are useful for counts such as arrivals, defects or incidents when the average rate is reasonably stable.
Frequently Asked Questions
Can lambda be negative?
No. The average event rate must be zero or positive.
Are events assumed independent?
Yes, the standard model assumes independent events.
What if the rate changes over time?
A single-rate Poisson model may then be inappropriate.
Methodology, Assumptions and Interpretation
A Poisson model gives P(X=k)=e^(−λ)λ^k/k! for counts in a fixed interval when events occur independently at a constant average rate. Overdispersion, clustering or a changing rate can make the model inappropriate.