Poisson Distribution Calculator

The Poisson distribution describes the likelihood of a given number of independent events occurring within a fixed interval — such as per hour, per square metre, or per page — when the average rate λ is known. It is particularly valuable for modelling rare or low-probability events: radioactive decay, server requests per second, customer arrivals per minute, or defects per unit of product. The key assumptions are that events occur independently, at a constant average rate, and two events cannot occur at precisely the same instant.

The probability mass function (PMF) is P(X = k) = (λ^k · e^(−λ)) / k!, where λ is the mean number of events per interval, k is the exact count of events you want to evaluate (a non-negative integer), e ≈ 2.71828 is Euler's number, and k! is the factorial of k. A remarkable property of the Poisson distribution is that its mean and variance are both equal to λ, so the standard deviation is √λ. The cumulative distribution function (CDF), P(X ≤ k), is the sum of PMF values from 0 to k, giving the probability of observing at most k events. The survival function P(X > k) = 1 − P(X ≤ k) gives the probability of observing more than k events.

Practically, the Poisson distribution is used wherever counts of arrivals, failures, or occurrences are needed. Call centres use it to plan staffing based on expected call volumes. Network engineers model packet arrivals and buffer overflows. Epidemiologists estimate disease incidence rates. Actuaries price insurance policies for accident frequency. Quality control engineers model defect rates on production lines. Wherever you have a count process with a stable rate and independent occurrences, the Poisson distribution provides an elegant and accurate probabilistic model.

The Poisson distribution rests on strict assumptions that may not hold in practice. First, events must be independent — if one event increases or decreases the probability of another (clustering or inhibition), Poisson is inappropriate; consider the Negative Binomial distribution for overdispersion (variance greater than mean). Second, the rate λ must be constant over the observation interval; if the rate changes over time (e.g., rush-hour traffic), a non-homogeneous Poisson process or piecewise model is needed. Third, k must be a non-negative integer; this calculator floors non-integer inputs to the nearest whole number. Fourth, for very large λ (typically λ > 20–30), the Normal approximation (μ = λ, σ = √λ) becomes accurate and computationally easier. Fifth, the Poisson distribution models event counts, not continuous measurements — do not use it for reaction times, distances, or durations (use Exponential or Gamma distributions for those). Finally, numerical precision may degrade for extremely large k or λ; this calculator uses log-space arithmetic to mitigate overflow, but values of k or λ above ~170 may still produce inaccurate results in standard floating-point arithmetic.