Derivation of the Poisson Distribution
A probability distribution may be modelled by a Poisson distribution if events occur randomly at a constant average rate,and each occurrence of each event is independent of every other occurrence. On top of this, events may never occur in pairs, so that two events occur at the same time.
If a type of event occurs at the ratethen in a small interval of lengthwe haveso that
Suppose events occur and obey a probability distributionas described above, thenis a Poisson distribution with parameter
Letdenote no events in the time intervalthen
Now prove by induction.
Assumeand use from above
Now multiply by the integrating factorto give
Integration now gives