Poisson Distribution Calculator
Compute P(X=k) and cumulative probability for a Poisson distribution.
Input Data
Average number of events per interval (>0).
Number of occurrences (non-negative integer).
Results
0.224042
0.42319
At a glance:P(X=k) = e⁻ᵏ·λᵏ / k!, where λ is the mean rate.
Formula
P(X = k) = e^(−λ) · λᵏ / k!
$$P(X = k) = \frac{e^{-\lambda}\lambda^{k}}{k!}$$How to Use
- Enter the average rate λ (e.g. 3 events per hour on average).
- Enter the observed count k (non-negative integer).
- The calculator outputs the probability of exactly k events P(X=k).
- It also gives the cumulative probability P(X ≤ k).
FAQ
How is Poisson related to the binomial?
When the number of trials n is large, the success probability p is small, and np ≈ λ is fixed, the binomial approaches the Poisson. Poisson is used for modelling rare events.
What are the mean and variance of a Poisson?
Both the mean and variance of a Poisson distribution equal λ. If your observed data show variance much larger than the mean, you may have overdispersion.
References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.