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Consider a more extreme case for a follow on to this question here about the Prussian soldier-face kicking example for Poisson random variables

Relating the binomial probability distribution to the Poisson Distribution in an example

Given an experiment to identify a probability p, for n trials, without prior knowledge of p... in cases where any where between 1, $10^2$, or $10^3$ over N samples may represent the success rate, and n observed events may range anywhere between $10^3$ - $10^8$. Events remain independent of other events throughout.

p may have true orders of magnitude of 0.005, 0.00001, but generally no less than $10^-8$, but there is a clearly an unknown order of magnitude to which p belongs.

Lets say I want to come up with a confidence level for p... and I do not know if I have sampled sufficiently in my experiment with N samples and observing n events.

  1. Does a binomial confidence interval using incomplete beta distribution apply (Clopper-Pearson CI), or Inv-Chi Square distribution better represent the uncertainty / dispersion of the true mean? (As both represented in Univariate Discrete Distributions by Johnson, Kemp, and Kotz)
  2. How does one know to assume a Poisson process, and that the sample number $n\rightarrow\infty$ applies for a general case, when it is unknown what the true number of N should be to approach the true probability $p= n/N$?

I recognize an estimator for p, is not the same as the true probability p as above I referred to them interchangeably.

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