Poisson Distribution Calculator
Solve common Poisson count-probability questions from an average rate λ and integer count k, with mean, variance and standard deviation shown.
Rate-based event counts need a different model from fixed-trial success counts
Poisson uses one average rate λ to model how many events occur in a fixed interval. That is a different starting point from a binomial model, which requires a fixed number of trials and a success probability on each trial.
Lynara evaluates the Poisson mass function and cumulative tails in log space with bounded inputs so large powers and factorials are not formed as unstable floating-point intermediates. The result still depends on whether a constant-rate independent-event model is appropriate.
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Formula
P(X = k) = e^(−λ) × λ^k ÷ k!; mean = variance = λ
Example
For λ = 4 and k = 2, P(X = 2) is about 14.65% and P(X ≤ 2) is about 23.81%.
References for this method
Useful for
- Poisson probability homework
- Exactly / at most / at least event counts
- Queue, arrival and defect-count models
Questions
When is Poisson different from binomial?
Poisson models counts from an average rate over an interval, while binomial starts from a fixed number of trials with a fixed success probability.
What are the Poisson mean and variance?
For a Poisson distribution both the mean and variance equal λ, so the standard deviation is √λ.