### Everything points to Poisson

#### (Jan 16, 2018)

One recurring theme in our forthcoming book, Modelling Mortality with Actuarial Applications, is the all-pervading role of likelihoods that suggest the lurking presence of a Poisson distribution.  A popular assumption in modelling hazard rates is that the number of deaths observed at any given age is a Poisson random variable, so perhaps that might explain it?  Surprisingly, it is the other way round - it is the very nature of the data in a survival model that leads inexorably to the Poisson distribution, even if we assume no such thing.

Stripped back to basics, we observe $$n$$ individuals and record our observations as:

• The length of time $$E_i$$ that the $$i$$th person was observed and alive; and
• An indicator…