I have results in the form {elapsed time, number of individuals}. This might be considered a Poisson point process. Using Mathematica how do I check if my results follow a Poisson distribution with the mean count dependent on the length of the elapsed time?


This might be useful:

h = DistributionFitTest[x, PoissonDistribution[Mean@x], "HypothesisTestData"];
h["TestDataTable", All]
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In another question you posted some data:

data = {{0.12, 10}, {0.34, 11}, {0.50, 10}, {0.66, 9}, {0.99, 10},
        {1.08, 9}, {1.44, 8}, {1.66, 9}, {1.68, 10}, {2.00, 11}}

where each pair represents {time elapsed, count}.

For this data and what you've described a reasonable first attempt of a model might be that the data is generated from a Poisson process where the counts follow a Poisson distribution with mean $t \lambda$ where $t$ is the elapsed time and $\lambda$ is the expected number of counts per unit of time.

Because there is variability in the elapsed times, the variance of the counts is expected to higher than a Poisson distribution with a constant mean. For a Poisson distribution the variance is equal to the mean.

If for the moment we ignore the elapsed time and just look at the counts we find the mean and variance are estimated with

N[Mean[data[[All, 2]]]]
(* 9.7 *)
N[Variance[data[[All, 2]]]]
(* 0.9 *)

That the variance is so much smaller than the mean (in fact around 1/10th of the mean) means that the variability is smaller than what is expected for a Poisson distribution. So the hypothesis of a Poisson distribution (with or without accounting for elapsed timd) is untenable. (One doesn't always need a formal P-value to make decisions.)

Now if the data supplied is not the actual data for which you need assistance, please supply that data so that a better answer can be given. (If the data does follow a Poisson distribution after accounting for elapsed time, then using GeneralizedLinearModelFit with a LinearOffsetFunction might be what you need.)

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