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What I want to know is, how does one go about finding the distribution that fits a particular set of data? I searched, but found nothing really satisfactory.

Do we have any historical anecdotes which can shed some light on this? For example, I've read a very brief account of how de Moivre discovered the normal distribution, but it doesn't really get down to details that make it possible to follow his thinking. Similarly with other accounts that I ferreted out.

I ask because I was looking at the number of open option contracts for AAPL today, and it occurred to me that there has to be some way to go about fitting a distribution to the data, but I don't recall ever seeing this in any of my reading or any of the courses I took back in the day.

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Your question is very broad. Just to check, though, do you want to know something about probability distributions or do you just want to know how to find a model that fits some data? – Thomas Dec 6 '12 at 2:52
If it is too broad, then how about the particular situation I mentioned, which was the number of open AAPL option contracts? – Zippy The Pinhead Dec 6 '12 at 3:52
Do you want to know something about probability distributions or do you just want to know how to find a model that fits some data? – Thomas Dec 6 '12 at 13:39
In this case, I think the latter would be sufficient. – Zippy The Pinhead Dec 7 '12 at 2:06

Since no one else have tried to answer your question I will give it a go even though I by no means am an expert in what I think that you are asking about.

It is quite the problem in math to come up with an equation that models some given data. And it sounds like you would like to know how to come up with a model/function that that models the options for Apple. And it makes sense to do this since you could try to use the model to predict what will happen in the future. However, it is mathematically impossible (at least practically) to actually do this. You might do research on this yourself, but the stock market (for example) doesn't just follow a nice pre-determined mathematical model. Lots of more of less random events have huge impacts on the markets.

However, you of course can make mathematical models in math. If you for example consider some data $x$ and $y$ and you plot the data in an $xy$-coordinate system, and if the points appear to be on a straight line, then there are "simple" ways to find the linear function that "best" describes the data. When we do this we call it linear regression.

Now if the points appear to follow some other pattern, then there are other models that you might use.

You might also have some theory based on some principals that gives you some equations that ought to describe a behavior. And so you can try to find the equation (of that given form) that best describes the data.

However, when it comes to the financial markets, I don't know enough to say much useful, but from what I understand this is all current research. You might benefit from asking an economist. The key thing, I would say, is to come up with a "type" or "form" of equation that you can try and fit your data to.

So for example I might have reasons to believe that $y$ should depend on $x$ like $y = ax^2 + bx + c$, and so the problem is: how do I find the coefficients $a$, $b$, and $c$?

Now it is a fact, that given a finite number of points, you can always find a polynomial of high enough degree, such that the polynomials describes the points perfectly. But then usually if you add one more point, then the polynomial doesn't work (well) anymore.

I hope this helped a tiny bit... again, your question is very broad and probably should be closed. However, since you wanted to know a bit about models and data, I thought that I would offer this bit.

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