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Why is poisson distribution always studied as time interval based when it is just a special case of binomial distribution?

Say I have a machine producing pins. X= perfect pin produced (success event) P(perfect pin) = 1/10000 N=20000 Find P(X=3) that is probability 2 perfect pins are produced in 20000 trials Now binomial cannot be applied given large n and small p

But I can model X~P(np) Using E(X) for lambda

Why and how does the time interval comes in play?

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The binomial distribution can always be applied to model a binomial process, even with a large n and a small p. The normal approximation for a binomial distribution cannot, but that was not the question unless the product of j and o is sufficiently high.

More importantly: The Poisson distribution is not a special case of the binomial distribution. It is a different distribution used to model a different process. I’ll explain using meteors and meteorites.

The Poisson distribution models the number of events that occur in a given time period. For instance, the number of meteorite strikes on the Earth in a year. In this case, there is no limit to the number you are modeling (you can always have more meteorite strikes). The number you are modeling is also discrete (you can’t have 1.5 meteorite strikes).

The binomial distribution models the number of success in a sequence of independent events. For instance, the number of meteors that became meteorites. In this case, the limit to the number of meteorites is also the number of meteors. The number you’re modeling is discrete, like the Poisson distribution.

In the Poisson distribution, the “time interval” doesn’t come into play. It is the defining factor of the distribution.

Poisson and Binomial are two different distributions for two different processes. Understanding these differences fully requires a top-down fundamental understanding of both processes that can only be obtained through extensive reading and mathematical work.

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  • $\begingroup$ thanks a lot!!. Yes I correct my statement that binomial cannot be applied to it is very difficult to compute. The book am referring had something written along those lines and that the binomial distribution is approximated to poisson distribution in such cases. Nonetheless thanks for explaining the difference. Could you please provide an example for how a binomial can be treated as poisson and how we can introduce the idea of time in such cases Say for the example i quoted in question $\endgroup$ – user1673216 Jul 1 '19 at 15:45
  • $\begingroup$ 1) The Poisson distribution is occasionally approximated by the Binomial distribution, but the Binomial Distribution is never approximated by the Poisson Distribution. 2) A binomial distribution is never treated as a Poisson distribution and time is never a factor in a binomial distribution. The example you provided in the question is a binomial process and not, at all, a Poisson process. $\endgroup$ – Matthew Anderson Jul 1 '19 at 19:42
  • $\begingroup$ i appreciate all the help you are providing me. But could you please loom at the below link. They derived the poisson distribution from binomal and define two types of intervals which can be there in poisson distribution.. one is time interval and other space interval source: newonlinecourses.science.psu.edu/stat414/node/85 $\endgroup$ – user1673216 Jul 2 '19 at 9:17
  • $\begingroup$ Everything I have stated so far is correct. The appended source states that for some values of n and p, it is a difficult computation to find the binomial probability. It is difficult, but it is correct. The Poisson distribution and the Binomial distribution are behaviorally similar (although not the same and used to model different processes). Because of their similarity, the author of the appended source shows that it is possible to roughly approximate the Binomial distribution with the Poisson, although I would tell you to avoid that as much as possible. $\endgroup$ – Matthew Anderson Jul 3 '19 at 0:56
  • $\begingroup$ Thanks mate!! ! $\endgroup$ – user1673216 Jul 3 '19 at 6:01

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