I have data set from real life (number of emails send by one user during two year period - some days are without sent emails) since this is discrete event my focus is on Poisson. Since variance is not small (some days are with one email and another is with let's say 30 emails) and it is not equal to mean (some of you will say that it is not Poisson if this condition is not fulfill), I want to test it.
I need some guidelines on how to do this. My idea is to generate, using R, some data set with $n$ elements, using Poisson distribution and then using some statistical test (maybe $\chi^2$ to compare my data with generated one) to find if they are similar to prove if my distribution is Poisson or not.
If this is not a correct road map, I will be thankful for guidelines. What I plan to do is to simulate user email generation process in order to predict future behavior.