Calculadora de Distribuição de Poisson
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
P(X=k) = e^(−λ) · λᵏ / k!
About this calculator
The Poisson Distribution Calculator computes the probability of a specific number of events occurring in a fixed interval of time or space, assuming events occur at a constant average rate and independently of time since the last event. The formula used is P(X = k) = e⁻λ · λᵏ / k!, where λ is the average rate and k is the desired number of events. This tool is useful for modeling phenomena such as number of phone calls per hour, traffic accidents per day, or defects in a piece of fabric.
To use the calculator, enter the value of λ (mean) and k (number of events). The result is the exact probability of exactly k events occurring. The Poisson distribution is widely applied in fields like traffic engineering, quality control, biology, and finance. For example, a call center manager can estimate the probability of receiving 10 calls in an hour, knowing the average is 8 calls per hour.
Cautions: The Poisson distribution assumes events are independent and the rate λ is constant. It is not suitable for events occurring in very short intervals or when there is temporal dependence. Also, ensure λ is a positive number. For large λ (above 20), the normal approximation may be more practical. This calculator provides accurate results for λ up to 100 and k up to 170, due to computational limits.
Frequently asked questions
What does the parameter λ mean in the Poisson distribution?
λ is the average rate of occurrence of the event in the considered interval. For example, if on average 5 accidents occur per day, λ = 5.
Can I use this calculator for events with a very high average rate, like λ = 1000?
No, the calculator works for λ up to 100. For larger values, use a normal approximation or another tool.
What is the difference between the Poisson distribution and the binomial distribution?
The binomial models the number of successes in a fixed number of trials, while the Poisson models the number of events in a continuous interval, with no upper limit.
Does the calculator show the cumulative probability (P(X ≤ k))?
No, this calculator provides only the exact probability P(X = k). For cumulative probabilities, you need to sum individual probabilities.
What should I do if the result is 0?
If the result is 0, it may be due to a very large k for the given λ. The probability of events far above the average is extremely small.