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 length
we have
so that
Suppose events occur and obey a probability distributionas described above, then
is a Poisson distribution with parameter
Letdenote no events in the time interval
then
Therefore
Letto give
Integration gives
so that
In general
So
Aswe obtain
Now prove by induction.
Assumeand use from above
as
Now multiply by the integrating factorto give
Integration now gives