# When do you know when modeling something with a binomial distribution is better than a Poisson?

I was trying to understand when its better to model some random variable as being distributed by a Poisson Distribution or when being modeled as a Binomial Distribution.

I was reading the following article:

http://www.umass.edu/wsp/resources/poisson/

but was not sure if I appreciated it.

What kind of structure does your data need, for it to be better approximated by a Poisson rather than a Binomial? What assumptions are necessary to separate when to model by Poisson or by Binomial?

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The binomial models "the number of successes in n trials". A binomial distribution has a theoretical upper bound ... the parameter n. The Poisson is open ended. It is typically used to model rare events within a certain time period, but there is a non-zero probability attached to every possible number of outcomes.

The Poisson distribution is actually a limiting case of the binomial: if you let the number trials go to infinity and the probability of success go to zero in such a way that $np=\lambda$, you get a Poisson.

EDIT: in response to the comment below, let me clarify what is meant by the limit.

Let's take the classic Poisson example: number of deaths per year due to horse kicks in a XIXth century Prussian cavalry regiment. Let's suppose the mean number is 2. I could model this as a binomial in the following way: divide the year into 365 days. Posit that the probability of a death on a particular days is 2/365 (and assume we have at most 1 death per day). Then use a binomial distribution to calculate the probabilities of 0, 1, 2, 3, etc. deaths in 365 days. If you do the math, you will get almost the same result as if you calculate the Poisson probabilities for 0,1,2,3, using a Poisson with mean 2.

What you will find, however, is that working out the binomials takes a lot more time -- at least, if you are doing it by hand .. and involves calculating a lot of pretty horrible combinatorial coefficients. The Poisson is simpler to calculate, which is one reason it was preferred for rare events, back in the day when one needed to do these things by hand.

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Sorry if this is very obvious to you but, what do you mean by "acting as a limiting case"? Do you mind clarifying or explaining that just in a little more detail? Thanks! :) –  Charlie Parker May 15 at 1:23
I changed my answer to respond to your question. –  Placidia May 15 at 1:30