The Poisson distribution models the number of independent events occurring in a fixed interval of time or space, given a known average rate λ — the number of customer arrivals per hour, defects per meter of cable, typos per page, or emergency-room admissions per night. It is fully specified by the single parameter λ (the mean rate), and — notably — its variance always equals its mean, a distinctive property called equidispersion. Like the binomial, the Poisson is a discrete distribution (a PMF): X can only take whole-number values 0, 1, 2, 3, …, with no upper bound.
Frequently Asked Questions
How is the Poisson distribution related to the binomial?
The Poisson distribution is the limiting case of the binomial when n → ∞ and p → 0 while np stays fixed at λ — that is, a very large number of trials, each individually very unlikely to succeed, but with a stable overall rate. This is why Poisson is the natural choice for 'rare event' counts (accidents, mutations, defects) rather than a fixed, moderate number of trials.
What does it mean that the variance equals the mean?
This property, called equidispersion, is a strong assumption: if your real count data are more spread out than their mean (overdispersion — common when events cluster, e.g. one customer complaint triggering several related ones), a plain Poisson model will understate the true variability and give overconfident (too-narrow) confidence intervals. MindStat's count-regression tools include negative-binomial models specifically to handle overdispersed count data.
Can λ be a decimal, like 2.5 events per hour?
Yes — λ is an average RATE and is very often not a whole number (e.g. 2.5 calls per minute, averaged over many minutes). Only the outcome k (the actual count in a specific interval) must be a whole number, since you cannot observe 2.5 discrete events. This calculator accepts any positive λ, including decimals.