(adsbygoogle = window.adsbygoogle || []).push({}); Are both sensible (equivalent? contradictory?) interpretations of "Poisson" behavior?

I've come across two quite distinct notions (or so it seems to me, anyway) of Poisson behavior and I'm not sure if they're equally sensible or perhaps even equivalent. I'll apply both "views" to the same case to show you what I mean.

The first view is how I first met Poisson in my statistics text book (and the way I did not like):

Say we have n houses, and we know that the probability of one house burning down in the course of one year is p. We can assume that n is very large and p is quite small. We're interested in knowing the probability ofthe number of houses that burn down in the course of one year. If we look more closely we see, as the burning down of every house is a Bernoulli experiment, that this follows a binomial distribution with variables n and p. As n is large and p is small, we can approximate this distribution by a Poisson distribution with parameter [itex]\lambda = np[/itex] (with [itex]\lambda[/itex] a "normal" size, way larger than p, way smaller than n).

The second "Poisson" view is called aPoisson process(it has got its own wiki page):

Clear your head of the previous case. We now regard the burning down of one specific house as an intrinsically random event, hence the time until burning down is modelled well by an exponential distribution. Call the exponential distribution parameter [itex]\lambda[/itex]. We want to know the probability distribution for the number of houses that burn down after a time t (afterwards we will take "t = one year"). As the probability of a house burning down in a given time interval dt is [itex]\lambda \mathrm d t[/itex] (you can see this as a consequence of the memorylessness of the exponential distr.), we can see that "the number of houses that burn down after a time t" as a sum of [itex]\frac{t}{\mathrm dt}[/itex] bernouilli experiments, each with probability [itex]\lambda \mathrm d t[/itex]. In the limit [itex]\mathrm d t \to 0[/itex], this is described by a Poisson distribution with parameter [itex]\lim_{\mathrm d t \to 0} \left(\lambda \mathrm dt \right) \left( \frac{t}{\mathrm d t} \right) = \lambda t[/itex].

You see that in both cases we arrive at a Poisson distribution but in quite distinct ways. (For ease, take the units of lambda to be "per year", then the lambda in both cases is equal.) But is the result equivalent? I think not, right? For one thing, in the former case, the end result depended on the number of houses, whereas in the latter case it didn't enter at all.

Even conceptually it is quite different, no? In the former case, it was truly a Binomial distribution which we mathematically approximated by a Poisson distribution (cause n was big and p was small), but in the latter case it was the Binomial distribution which was a temporary approximation, but by taking the limit we got the actual nature of the problem. But perhaps this note is too much philosophy and too little hardcore mathematical objection.

Anyway, I was wondering, are both sensible applications of the Poisson distribution? I think both derivations make sense, but on the other hand the results are different. Which of the two should a company use? Or should we expect them to make similar predictions? Is there any of the two more true than the other? Do they apply in distinct cases?And am I the only one who thinks the second view is somehow more pleasing?(I just encountered the second view in my physics Markov course, and only now do I feel I somehow understand Poisson behavior.)

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# Are both sensible (equivalent? contradictory?) interpretations of Poisson behavior?

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