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I work on empirical distributions generated by a Monte Carlo. Some of these distributions are close to Gaussian, other looks more like Multinomials and still others like uniforms.My problem is to find a common value for representing the "central case". For Gaussian and uniforms, I can use the E[], but for Multinomials, the E[] will just lie in the middle of nowhere. In this case, the Mode() would be more useful.Of course, the distributions being empirical are not really Gaussian, Multinomial or Uniform, so I don't know which measure to apply a priori.Does someone had had similar issues and how did you solve it?Many thanks

How can you generate numbers from monte carlo without knowing the distribution ?

I simulate f(X) with X multi variate Gaussians and f() highly non linear.The distribution I am talking about is the distribution of f().I hope I am clear?

I would recommend using the median

I thought about it also, but the median can also be an infrequent or impossible value.It seems I need to be able to characterize the distribution first (perhaps by comparing the expected value/Mode/Median) before I can choose which value corresponds better to the "most likely" value that I can then use as a forecast.

QuoteOriginally posted by: regis99I thought about it also, but the median can also be an infrequent or impossible value.So can expected value. Why is this a problem?

If you want to use the "most likely" value as a forecast (as opposed to pricing contingent claims for example), one desirable property would be to have a value that corresponds to something "real".For instance, let's say that I have a population with 60% measuring 1.60m and 40% measuring 1.80m.Although the mean is 1.68, as a forecast of the most likely height, you would rather return 1.60m (=the mode).But depending on the shape of the distribution, the mode is not always the best value to return, hence my problem.

The core of problem is that you have a conflicted definition of the 1st moment.If you really want the "most likely" value, then the answer is the mode. If you want an answer that is "central" in events -- defined by half the events being higher and half being lower -- then the answer is the median. If you want an answer that is "central" in error -- defined by half the mass of errors WRT the center and the events being higher and half being lower -- then the answer is the mean. If you like the central-in-error definition but want some robustness to extreme value outliers, then the answer might be trimmed means.I'd think carefully about the downstream application of the first moment and how the values created from the different definitions propagate. For example, if the events represent valuations or returns on some instrument and those values accumulate to form a portfolio value or profit-and-loss, then the mean will be a better representation.

Thank you. You helped me get a better understanding of the problem.Now I "only" have to find a solution, but I least my ideas are clearer.