Examples of processes that are not Poisson?

I am looking for some good examples of situations that are ill-suited to model with a Poisson distribution, to help me explain the Poisson distribution to students.

One commonly uses the number of customers arriving at a store in a time interval as an example that can be modeled by a Poisson distribution. I’m looking for a counterexample in a similar vein, i.e., a situation that can be regarded as a positive count process in continuous time which is clearly not Poisson.

The situation should ideally be as simple and straightforward as possible, in order to make it easy for students to grasp and remember.


Number of cigarettes smoked in a period of time: this requires a zero-inflated process (e.g. zero-inflated Poisson or zero-inflated negative binomial) because not everyone smokes cigarettes.

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