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I'm looking for a tool that does "probability distribution fitting" given a set of data points. Sort of like curve fitting, but tries to fit to standard density distributions.

For example if I input 

(0, 0.0497871), (1, 0.149361), (2, 0.224042), (3, 0.224042), (4,0.168031), (5, 0.100819), (6, 0.0504094)

I would hope that it would tell me these data points fit a Poisson distribution.

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    $\begingroup$ This needs clarification: is your input supposed to be independent samples from the distribution, or points sampled from a graph of the density, or something else? Basically, what do you mean by "sample"? $\endgroup$
    – Darsh Ranjan
    Commented Oct 21, 2009 at 2:13
  • $\begingroup$ The sample data is the probability distribution function of a Poisson variable with mean 3. So it seems that the OP wants to give a pdf and find out what distribution it is. $\endgroup$
    – Michael Lugo
    Commented Oct 21, 2009 at 2:22

5 Answers 5

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It appears to me that you want to perform a goodness of fit test. What this test allows you to do is compare your sample data to the poisson distribution with a certain parameter via a statistical hypothesis test. Check out the link for more information wikipedia.

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    $\begingroup$ This person might want something more, like a test that tells you whether a Poisson distribution, or Gaussian, or binomial, or something else is better. Given some explicit class of cases to check (and a guarantee that the chosen distribution actually is one of them) this sort of statistical tool will probably work. But the general problem is probably too open-ended to have a mathematical solution. $\endgroup$
    – Kenny Easwaran
    Commented Oct 21, 2009 at 5:45
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Google suggests there are tools.

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You need to have some candidate distribution before you do a goodness of fit test.

If you suspect your data follow a Poisson distribution, I'd start by computing the sample mean and variance. If these are equal, maybe you do have Poisson data. If your sample variance is appreciably larger than your sample mean, you might try negative binomial next.

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The following document might be of some use:

http://cran.r-project.org/doc/contrib/Ricci-distributions-en.pdf

Depending on whether you use R, it might be of a variable level of usefulness. The general approach still stands though.

As a previous commenter has said, this isn't really a well-posed problem and does not have some closed mathematical solution. However, you can harness some computational tools to aide you in this. A good starting point is always plotting the data to see what they look like, you can then hypothesize a distribution you think could fit and then finally test that hypothesis via a goodness of fit test (as suggested by another commenter.)

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there is a software called @risk ( www.palisade.com) which can fit all the possible distributions and then order the distributions based on goodness of fit

Hope this helps

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