the probability mass function of X:
Homogeneous Poisson Process
This process is characterized by a rate parameter λ, also known as intensity, such that the number of events in time interval (t, t + τ] follows a Poisson distribution with associated parameter λτ.
Non-homogenous Poisson Process
If the rate λ changes over time (being λ(t)) a poisson process is said to be non-homogeneous. The expected number of events between time a and time b is as λ_a,b: the number of arrivals in the time interval (a, b], given as N(b) − N(a), follows a Poisson distribution with associated parameter λa,bcounts of the number of "events" inside each of a number of non-overlapping finite sub-regions should each have a Poisson distribution and should be independent of each other.