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Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking into this subject but I couldn't find this subject in numerical analysis. I was interested in this for approximating expected values, standard deviations, etc.

Maybe someone has encountered this subject before. Thank you.

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closed as off-topic by Joonas Ilmavirta, Stefan Kohl, Alex Degtyarev, Ricardo Andrade, Suvrit Jun 7 '15 at 12:41

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Joonas Ilmavirta, Stefan Kohl, Ricardo Andrade, Suvrit
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ What are "discrete data points"? Random samples? Or values of the pdf at some prescribed points? $\endgroup$ – Federico Poloni Jun 7 '15 at 7:42
  • $\begingroup$ Just a random sample. Can you numerically create a discrete pdf from a large enough sample? $\endgroup$ – user136693 Jun 7 '15 at 8:17
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    $\begingroup$ If you want some better answer than the one given already, we really need more information. But, anyhow, you should consider asking that at Cross Validated instead, were it will receive more interest than here. stats.stackexchange.com $\endgroup$ – kjetil b halvorsen Jun 7 '15 at 10:54
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IANAS, but I'll try an answer.

The problem is known as density estimation. A first option (which you probably will not find satisfactory) is an average of Kronecker deltas centered at the sampled points. Or you can replace the deltas with Gaussians or other shapes (kernel estimation).

As far as I know, in practice it is more common to assume a fixed (parameter-dependent) distribution and fit its parameters to the observed points with techniques such as maximum likelihood.

If you just need to find mean and variance, sample estimators are the traditional choice, and they do not go through the distribution. Their statistical properties are well studied.

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  • $\begingroup$ This is exactly what i needed to hear. Thank you for the thoughtful response $\endgroup$ – user136693 Jun 8 '15 at 6:16

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