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I am dealing with a Monte-Carlo simulation which provides me with a value $x$ as a result. I can run the simulation $n$ times, which gives me $n$ values. I don't know in advance what the probability distribution function of the output values is. What I want to be able to do is to place an uncertainty on the sum or the mean of these values, but I need the uncertainty to be sensible even for small values of $n$.

For example, if I have $n=2$, and the two resulting values of $x$ are very similar (say 9.9 and 10.1), then intuitively, I know that the mean is still not necessarily accurate, because there are only two samples (whereas a standard error in the mean would give me a small uncertainty, because it uses the standard deviation).

Is there a formal way to deal with this kind of situation? In a way, I'm looking for something analogous to Poisson statistics, which allows one to place an uncertainty even on a single value, but in a more generalized way.

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If you have any guess on the probability distribution of the result, you can either: (1) find the best parameters that describe your distribution, and estimate the uncertainty from there; or (2) use some prior on the probability distribution and proceed Bayesically. The simplest example is assuming that everything is normal, estimating the standard deviation, and concluding the approximate distribution of the mean. – Yuval Filmus Jul 31 '10 at 23:58
Could you provide more information? As you have described the problem, the simulation could even fail to terminate for some inputs, which makes problems of bounding uncertainty really hard. – András Salamon Aug 1 '10 at 2:37

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