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Essentially, I am asking if it is possible to parameterize an empirical distribution given a few observation of the distribution at different means.

For example:

Suppose I have a discrete empirical distribution that looks like the following: enter image description here

It has a mean of 1.745. This distribution is the historical number of arrivals into a system (for which I am building a simulation model), and it doesn't match any popular parametric distributions (i.e. chi-squared tests for Poisson, Binomial, etc come back poorly). Not only this, but using a Poisson arrival process is not valid because time between arrivals is most definitely not exponential.

I have a client that will produce forecasts for the mean number of arrivals. Let's say they forecast mean number of arrivals to be 3.1 in 6 months. How should the histogram change, given that I know that at mean 1.745 it looked like above? Would that be helpful if I also know that at mean 0.519 it looked like X, at mean 1.11 it looked like Y, at mean 2.13 it looked like Z, etc? What if the forecasted value is beyond any of the means I currently have observed?

Perhaps the client could also produce both the mean and perhaps some confidence intervals for number of arrivals, giving a little bit more information. How could I incorporate that into the arrival distribution?

Is it worth it to continue with an empirical distribution like I currently have, or just go ahead and use a Poisson distribution for number of arrivals (even though the distribution doesn't match well).

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