Why this relationship works
A Poisson model assigns e⁻ˡambda λᵏ/k! to event counts when occurrences arise independently at a stable average rate.
For this page, poisson distribution is interpreted under the stated convention and input order.
Math calculator
Poisson Distribution Calculator
Find exact or cumulative event-count probability from an interval rate. The calculation trail makes the reported poisson probability easier to reproduce.
Poisson Distribution inputs
Values for this result
When the result is useful
It models arrivals, calls, defects per length, rare incidents, and counts across time or space.
Where the shortcut stops
The interval attached to λ must match the interval counted. Clustering, changing rates, or a hard maximum can make the model unsuitable. If the Poisson Distribution assumptions do not fit, consider distribution mean.
Compare the result with the worked example's scale before relying on the reported poisson probability.
A worked example
With λ=4.5, exactly three events has probability about .1687. This Poisson Distribution example can be compared with fixed trials.
Building Poisson Distribution
Apply the mass formula for an exact count; sum from zero for a lower tail or complement counts below k for an upper tail.
From calculation to usable answer
Source values may arrive in a different order from the form. Map them explicitly to probability type, expected events λ and event count k, normalize units, and retain enough precision for the next step after poisson probability.
How the output responds
Mean and variance both equal λ, so increasing λ moves typical counts upward and spreads them. Watching this response separates a data-entry mistake from an unexpected but valid value.
Precision and reporting
If probability type, expected events λ and event count k are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
Binomial specifies a fixed number of opportunities; Poisson does not. This page deliberately reports only the former interpretation.
Label the output as poisson probability in notes or tables. Store probability type, expected events λ and event count k beside it when the result will be reused in a later stage.
Assumptions behind Poisson Distribution
The Poisson Distribution formula describes a particular experiment built from Probability type, Expected events λ, Event count k. Write down whether the trial count is fixed, objects return after selection, events overlap, or outcomes are equally likely. Similar-looking numerical inputs can require different formulas when one assumption changes.
For Poisson Distribution, after calculation, translate the answer back into a sentence about the original event. That wording should distinguish exactly, at most, at least, all, and none; substituting one of those phrases for another changes the event rather than its formatting.
Questions about Poisson Distribution
What is λ?
Expected events in the interval.
Can λ be negative?
No.
What is the variance?
λ.
Does at least include k?
Yes.
When does Poisson approximate binomial?
For large n and small p.