Poisson distribution for rare event

Question

I was taught in school that Poisson distribution is usually used to model rare events. And I understand the Poisson process is such that the probability of an event in one interval is independent of another interval and the probability depends on the length of the interval. However, I don't understand why an event has to be a rare one. Anyone knows? Thanks! =)

Answer 1

For a Poisson process $X\sim P(\lambda)$:

• the increments are independant with the same distribution
• the average increment of the process in an interval of size $\Delta t$ is $E[X_{t_0+\Delta t} - X_{t_0}] = \lambda\Delta t$, so the empirical average number of realizations on the whole time of the empirical data is a good way to choose the parameter of the Poisson process
• when a certain interval is likely to be contain a big increment, the modelization via a Poisson process (or any other Levy process) is not a good idea...