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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.

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    $\begingroup$ You might want to explore the negative binomial instead of the Poisson which deals with over-dispersion. If you have an excess of zeroes there are methods for that too so perhaps expand your post if you need more help. $\endgroup$
    – mdewey
    Commented Jan 26, 2017 at 9:23
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    $\begingroup$ "What I plan to do is to simulate user email generation process in order to predict future behavior." This is not sufficient information. How do you plan to predict? Surely a distribution (there are several that model under/overdispersion and/or zero-inflation) alone is not a sufficient model. $\endgroup$
    – Roland
    Commented Jan 26, 2017 at 9:28
  • $\begingroup$ Do I need to include days without any emails (zero days) in data set? In simulation software I am using you can make custom behavior of user, you need to choose distribution (poisson or negative binomial) and inter_arrival time (exponentials in most cases). $\endgroup$
    – explorer
    Commented Jan 26, 2017 at 9:44

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You should definitely include days with zero emails in the dataset, those days are legitimate data. Omitting them you couldn't have a poisson distribution---the poisson have a positive probability for zero.

Also, what you have is time series data. So you should look into time series methods for count data, search this site. If you have enough data, maybe look into an latent variable process, where the poisson mean is a stochastic process with autocorrelation, and the daily count is then independent Poisson given the value of the latent process.

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