# 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 '14 at 1:23
I changed my answer to respond to your question. – Placidia May 15 '14 at 1:30

Traditionally, it is considered that data that fit a Poisson distribution best come from a Poisson process, e.g., a countable number of events during a certain time period. The example in the link you posted was a certain number of pieces of mail per day.

The Binomial distribution is the result of a Bernoulli process, e.g., the result of multiple trials with only two outcomes. A coin toss is the most boring but easy to understand example of a single Bernoulli trial. If you flipped the coin thousands of times you would approximate a Binomial distribution with a p=0.5, assuming your coin was fair (50/50 chance).

These two (the Poisson and Binomial) are indeed closely linked. Again, as your link pointed out, for relatively rare event Binomials, the Poisson distribution can substitute. Going back to the mail, if you considered every second or minute of the day a test of if you would get a piece of mail, then the vast majority of times you would fail this binomial test. (The p would be very low). You can instead approximate this with a Poisson. Of course, in this case, the Poisson is also more intuitive. A common use of the Poisson distribution is counting animals (in a forest, field, etc). You count a certain number of animals per area or time period. You could again, theoretically set up each minute as a test of if you'd find an animal or not (binomial), but no one would ever want to this.

So, aside from using my examples as a kind of heuristic, I guess it would be important to ask yourself:

1.) How common is a success? 2.) Is your time period continuous or discrete? This does the most to determine how "common" your success is, since when you coin flip you're not counting all the times you're not actively flipping as "failure". The clock starts and stops only when you're doing a single trial.

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