The Time Until the Next Occurrence in a Poisson Process
The Poison process is scalable so that if a random variable
is Poisson with occurrences at the rate of
in each time period
and we can write
then events occur in each time period
at the rate
and we can also write
Consider a Poisson process, and consider only that there are no occurrences in a time
so that
This is equal to
The distribution function of
the time to the first occurrence is then