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Say I have an experiment to determine the value of a certain parameter, $k$. I run the experiment once and I recover a probability distribution function for $k$. If I run is 20 times I get 20 distributions. How can I calculate the total probability distribution over all 20? Simply multiplying the pdf's together won't work, since even if the integral over a given pdf = 1, the integral over the product of all the pdf's need not be 1.

To clarify: assume, for example, that I have 20 different masses and one spring. For each mass, I hang the mass from the spring and determine a pdf for the spring constant (this is a contrived example, I know). Is there a way for me to then get the pdf of the spring that combined the data from all 20 experiments / masses?

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    $\begingroup$ Why wouldn't you just combine the data and fit one distribution. $\endgroup$
    – Michael R. Chernick
    Commented Aug 18, 2017 at 23:49
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    $\begingroup$ If you run it once, 20, ... infinite times doesn't it coincide to be the Expected value of your distribution? According to the Law of Large Numbers $\endgroup$
    – DarkCygnus
    Commented Aug 19, 2017 at 0:06
  • $\begingroup$ Edited the original post to clarify my situation! $\endgroup$
    – rrose
    Commented Aug 19, 2017 at 7:53

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You might be trying to "pool" the means and/or variances of your distributions. I've found multiple methods for pooling one or the other, but the following is the only resource I've found that pools both.

https://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/Pooled.htm

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