Probability · Distributions
Poisson Distribution Calculator
The Poisson distribution counts how many times a rare, random event happens in a fixed interval — calls to a switchboard in an hour, typos on a page, decays per second — when the events occur independently at a constant average rate λ. Set the rate and a count k and get every probability at once: exactly k, fewer than, at most, at least and more than — with the distribution drawn, the curious fact that its mean and variance are both λ, and an honest note on when real data break that promise.
λ is the mean number of events you expect in the interval (it need not be a whole number); k is the count you are asking about. All five probabilities are listed below — the chip just chooses which to headline and shade.
Result
In plain English
Some things happen at a steady average rate but at unpredictable moments — buses arriving, emails landing, raindrops on a tile. If each event is independent of the others and the rate does not change, the count over a fixed window follows the Poisson distribution. One number, λ, fixes everything: it is the average count you expect, and — unusually — it is also the variance, so the spread is locked to the mean.
- Poisson distribution
- The distribution of the number of independent events in a fixed interval when they occur at a constant average rate λ. Written X ~ Poisson(λ); the count can be 0, 1, 2, … with no upper limit.
- P(X = k), the probability mass
- e^(−λ) · λᵏ ∕ k! — the chance of exactly k events. The e^(−λ) keeps the whole distribution summing to 1.
- mean = variance = λ
- The defining quirk: the expected count and its variance are the same number, so the standard deviation is √λ. Doubling the rate doubles the spread in variance terms.
- the rate, and the interval
- λ is rate × interval length. Two calls a minute over five minutes is λ = 10; the same rate over one minute is λ = 2. Scale the interval and you scale λ.
- limit of the binomial
- The Poisson is what the binomial becomes when there are very many trials, each very unlikely, with n·p settling on λ. Rare events among many opportunities.
- overdispersion
- Real counts often vary more than their mean (clumping, changing rates), breaking the mean = variance rule. When the variance clearly exceeds the mean, the negative binomial is the honest model, not the Poisson.
Frequently asked
How do you calculate a Poisson probability?
The probability of exactly k events when the average rate is λ is P(X = k) = e^(−λ) · λᵏ ∕ k!, where k! is the factorial of k and e ≈ 2.718. For λ = 3 and k = 2 that is e^(−3) · 3² ∕ 2! = 0.0498 · 9 ∕ 2 ≈ 0.224. For “at most k” or “at least k,” add the relevant individual probabilities — this calculator lists exactly-k, fewer-than, at-most, at-least and more-than together.
What is the difference between the Poisson and binomial distributions?
The binomial counts successes in a fixed number n of trials, each with probability p. The Poisson counts events in a fixed interval with no upper limit on how many can occur, governed by a rate λ. They meet in the limit: when n is large and p small with n·p = λ, the binomial is closely approximated by the Poisson. Use the binomial when there is a clear, countable number of trials; use the Poisson when events arrive continuously in time or space and you only know the average rate.
Why are the mean and variance of a Poisson distribution both λ?
It falls out of the mathematics — derive the expected value and the variance from the formula and both come to λ. The practical upshot is that a Poisson process has no free spread parameter: tell me the average count and I have told you the variability, σ = √λ. This is also a testable assumption: if you observe count data whose variance is much larger than its mean (overdispersion) or much smaller (underdispersion), the Poisson is the wrong model and you should reach for the negative binomial or another alternative.
When can I use the normal approximation to the Poisson?
As λ grows the Poisson becomes roughly symmetric and bell-shaped, and for λ of about 10 or more it is well approximated by a normal distribution with mean λ and variance λ (so standard deviation √λ), ideally with a continuity correction of ±½. For small λ the distribution is sharply right-skewed and the approximation is poor — use the exact Poisson probabilities, which is what this page computes regardless of λ.