Poisson Distribution Calculator

Formula and method

P(X=k) = e⁻λ λᵏ ÷ k!. Exact probabilities use log-gamma arithmetic. Cumulative and survival probabilities use complementary regularized-gamma forms selected to preserve practical lower and upper tails instead of relying only on subtraction from a rounded value.

Worked example

For λ = 3 and k = 2, P(X=2) is approximately 0.2240418077, P(X≤2) is approximately 0.4231900811, and P(X≥2) is approximately 0.8008517265.

Assumptions and boundaries

• The model describes counts over a fixed interval with a stable average rate and independent event occurrence under the Poisson assumptions.
• Lambda is nonnegative; event counts are nonnegative integers.
• Lambda and event-count inputs are limited to 100,000 for responsive browser-local calculation.
• Binomial Distribution Calculator models a fixed number of trials rather than an event rate.

Inputs and results stay in this browser and are not sent to analytics, storage, URLs, or network services.