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I don't get this thing... I know that binomial distribution is used to know the probability of a X v.a. that sounds like this: X = "the probability of having 4 blue balls doing 10 extraction from a chest containing 7 blue and 40 white", and I know that poisson distribution and binomial distribution are really similar for lim_(p*n)->0(F(x)) (when p*n are really small...). I'm reading everywhere that the distribution of poisson is used a lot to approximate a big binomial... but what's its real purpose? what's the cases in wich I must use poisson and not binomial? (beyond the approximation case).

thanks in advance.

Willie Wong
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user2993157
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    The Poisson distribution can be derived from the binomial distribution. The Poisson is nothing more than the limiting case of the Binomial where n is large and p is small. – Islands Nov 15 '13 at 21:41
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    I would like to point out that a distribution that can be well approximated by another one is not immediately useless. – M Turgeon Nov 15 '13 at 23:10
  • @MTurgeon but is Poisson distribution needless for small n*p values? – user2993157 Nov 15 '13 at 23:28
  • Short answer: no. Why would a distribution be useless? The Poisson and the binomial distributions model different situations (even if there are similarities). Why would one of them be useless? – M Turgeon Nov 17 '13 at 18:17
  • @MTurgeon I mean, there's a case in which Poisson is needed but not as an approximation of the Binomial? – user2993157 Nov 18 '13 at 10:06
  • Well, I see plenty. Here's one example: think of modeling the rate of some disease. The numerator of this rate is a Poisson random variable, definitely not a Binomial – M Turgeon Nov 19 '13 at 05:22

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The Poisson distribution is also called the "law of rare events" -- it is the distribution that counts the number of occurrences of an event given that the probability of the event is very small.

That should sound a lot like the binomial distribution. In fact, the Poisson distribution can be derived as a limit of the binomial distribution.

So what is the point of it? First, the fact that the limit exists gives us a lot of analytically useful results. The Poisson distribution is easier to work with than the binomial distribution. It is easier to compute the pdf and especially the cdf. Its generating functions have nice properties. Etc.

Second, in applications, the Poisson distribution serves in ways that the binomial distribution just cannot handle. Consider the case of radioactive decay. You're measuring the rate, using a Geiger counter, on a sample of hundreds of trillions of atoms. The binomial distribution is arguably applicable in this case, but are we really sure that atoms are "discrete" in the same way the integers are? (That is an empirical question, up to science to figure out) I would argue that without a priori knowledge, using the Poisson approximation is not an approximation of the binomial distribution. It is an approximation of the "real" behavior of radioactive decay, whatever that is.

nomen
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