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What I want to say is, there were some "indicators" that it could be a Poisson process, but the most important, since it is used to derive the Poisson distribution from a binomial distribution, is the fact that the events has to be "rare". In the demonstration, the limit applied to the binomial distribution consists in tend the "numer of trials" ##n## to infinity but at the same time to maintain constant the average number of successes ##\Lambda=np##, with ##p## the probability of success in each trial. That is, ##p## tends to 0. So here one could take the notion of what a rare event is. But there are a lot of cases where there is nothing that one can understand as a "trial". For example in a radioactive decay, there are no such trials. One can not say: "Wait a minute, this atom here maybe is trying to decay now! Let me check it. Oh no, it missed, it is still Uranium." So one can not compute a probability of success ##p##. One also can not design an experiment to generate an estimator of the quantity ##p##. So, finally, and I must apologize for the extension, my question is: How can I determinate if certain events that are happening are really a Poisson process? I know the main condition is that the events must occur independently, but where is the relation of that with the "rare" issue? For example in Drude's theory for metal conductivity, it works to model the electron collisions as a Poisson process, but occur millions of them a second, they are happening all the time. So they're not rare. It would seem that the most important fact is the "randomness" and not being rare after all. I mean, Poisson processes mathematically are points in a certain domain, for them to be frequent or rare is just a scale factor. But in the reality, maybe there is some kind of causality argument? Expecting that very distanced in time events should dissipate the influence of one on another? I mean, "very distanced in time" compared with the characteristics times of the processes generation mechanisms...

Just wondering, if somebody knows the real justification of the thing...

Thank you very much in advance.