Structure beneath the Poisson Distribution calculation
A Poisson model assigns e⁻ˡambda λᵏ/k! to event counts when occurrences arise independently at a stable average rate.
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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
Tasks that depend on Poisson Distribution
It models arrivals, calls, defects per length, rare incidents, and counts across time or space.
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.
With λ=4.5, exactly three events has probability about .1687. This Poisson Distribution example can be compared with fixed trials.
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.
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.
Reviewing the Poisson Distribution case
Use Probability type as the first Poisson Distribution checkpoint. Confirm Expected events λ, then anticipate Poisson probability. Repeat Poisson Distribution without reading the prior answer. If Event count k differs, preserve both Poisson Distribution versions and both values of Poisson probability.
Cross-checking Poisson Distribution
A boundary case can expose a Poisson Distribution setup error. Choose a simple Probability type, preserve Expected events λ, and predict Poisson probability. If the new Poisson probability conflicts with that prediction, inspect the Poisson Distribution entries before relying on the less convenient case.
Validating Poisson Distribution
Compare the noun in the Poisson Distribution question with the label Poisson probability. Recheck Probability type and Expected events λ if they differ. Correct arithmetic can still produce a related quantity instead of the intended Poisson Distribution output.
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.