StatReasonWhat this answers
The Poisson distribution answers "if events happen at this average rate, independently of each other, what is the probability of observing a specific count in this exposure window?" It is the standard model for rare, independent event counts: customer arrivals in an hour, defects per batch, calls per minute.
How it is calculated
Lambda, the expected count for your exposure, is your entered rate times your exposure. The exact probability of k events uses the Poisson probability mass function, computed via log-gamma for numerical stability at large lambda or k rather than the naive factorial formula, which overflows quickly. Cumulative probabilities use the same regularized incomplete gamma relationship this project's other distribution engines rely on.
Worked example
For a rate of 2 events per unit exposure and an exposure of 1 (so lambda=2): the probability of exactly 0 events is e^-2, approximately 13.5%, meaning even with an average of 2 events expected, there is still a meaningful chance of observing none at all.
Assumption audit
Common mistakes
A common mistake is leaving the exposure field at its default of 1 when the actual exposure window differs from the one used to estimate the rate, which silently understates or overstates lambda. Another is assuming independence when events actually cluster (a viral outbreak, a cascading system failure), which the Poisson model cannot represent and will systematically understate the true variability. A third is treating "rare" as a strict requirement; the Poisson model works for any nonnegative integer count, not only very small counts, as long as the underlying rate-and-independence assumptions hold.
Limitations
The Poisson model assumes a constant rate over the entire exposure and full independence between events; violations of either assumption (a rate that varies, or events that influence each other) mean the reported probability no longer matches the real process.