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I have a set of numbers (each one corresponding to a payment made from the same person) and I would like to assign a probability value to a new specified number given that historical data.

I've looked at the Chebyshev's inequality as a simple means to do that but I don't get the expected results. The formula evidences only obvious differences from the time series and also gives too little probability value to numbers that are lower than at least one of the historical ones.

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You have to construct an underlying model (even if you assume payments do be independent and follow the same distribution, you already build up a model) maybe estimate parameters of the model (if there any) from the history - and then the question can be answered. – Ilya May 25 '12 at 9:32
How could I do this parameter estimation? – angaran May 28 '12 at 12:51

Following on what Ilya commented, you can look at the history and try to model the distribution. Certainly you can take a mean and standard deviation and make a Gaussian. But there are many other things to think about. Maybe all payments are a multiple of $\$1 or \$100$. Do you expect that to continue? It seems that an arbitrary payment of $\$0$ is always possible-somebody can forget, but an arbitrary payment of $\$1$ is much less likely if all previous have been of order $\$1000$. The Gaussian distribution almost always has smaller tails than real-world cases.

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What could be a good strategy to build a model? I'm kinda new to this stuff.. – angaran May 28 '12 at 13:13
@angaran: there is no simple formula. In a sense, this is what science is all about. As I suggested, one choice is to use the observed mean and variance to make a Gaussian. You could also fit to something of the form $\frac 1{1+x^2}$. There could be details of the distribution of interest. You could start by plotting the data you have (you need lots to make any sense) and staring at it. – Ross Millikan May 28 '12 at 21:32
I plotted my data and it's not normal more similar to a Zipf distribution. Do you know a good source to learn about model fitting/parameter estimation? – angaran May 29 '12 at 10:37
@angaran: I like Numerical Recipes Obsolete versions are free. Any numerical analysis text should have a section. – Ross Millikan May 30 '12 at 7:03

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