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I am not very well versed in statistics so any clarification would be appreciated. I understand the mathematical derivation of Poisson from Binomial.

I can see just from plotting various Binomial distributions where I keep p constant and increase n, that Binomial will eventually converge to a Poisson.

That fits with the derivation where n approaches infinity, but I don't see why p has to be very small? What is the intuition behind p approaching 0?

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The idea is that the poisson distribution is a one-parameter distribution (often $\lambda$), and that $\lambda$ corresponds to $np$ in the binomial distribution. So the limit you want is for $n$ to get larger while $np$ stays fixed - and this is why $p$ has to get smaller. Both $\lambda$ and $np$ count the number of events, which is why the correspond in this way - and the limit is taken while keeping the mean number of events constant.

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So in a binomial distribution, p is the "true" probability of the trial (or as good as you can get with a large population size) and n is the number of trials you are conducting. In a poisson, there is no "true" probability, only the rate (lambda). Is n approach infinite and p approach 0 a good approximation for the true probability? – damiankao May 16 '13 at 10:43
What you need is to look at the limit as $n \to \infty$, and set $p=\cfrac {\lambda}n$. It is not just that $p \to 0$, but that this happens in a special way. This also ensures that the variance $npq \to \lambda$, as you will want if your limit is to have any chance of being a poisson distribution. – Mark Bennet May 16 '13 at 11:23

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