CDF of the ratio of Poisson and possibly-Poisson R.V.

[flux_factor]

Hello,

This is my first post - so let me know if I communicate incorrectly.

To start, note that my thread title may be misleading as to my actual problem. I think it describes my situation, but let me provide background and then restate my problem as I see it, so as to allow for a potentially different interpretation:

I have data that are integer frequency counts for J possibly-dependent populations over a common timeframe.

For each population, I currently assume observations are a sample from a Poisson-distributed random variable (I may allow different populations to follow different distributions such as a Negative Binomial in the future, but I'm only asking about the all-Poisson case in this post).

Main question: I want to find the CDF of the following random variable: one of the Poisson r.v.s divided by {the sum of all J Poisson r.v.'s divided by J}. I would type this in Latex, but I'm having real trouble getting it to show up correctly on the forum, even when referring to
https://www.physicsforums.com/showthread.php?t=8997

I think I can combine the individual moment generating functions and then take the inverse Laplace transformation to find the CDF? Is there another way (is there a simple analytical solution that I'm overlooking)?

Does the problem simplify if the random variables are assumed independent?

Thank you very much!

 

[mmwave]

Thank you for your post. Your question is an interesting one and I will do my best to provide a helpful response.

First, I would like to clarify that the use of Poisson distribution for your data seems appropriate, as it is commonly used for count data and can handle over-dispersion (which seems to be your concern with potentially using a Negative Binomial distribution).

To answer your main question, finding the CDF of the random variable you described can be done using the method you suggested - combining individual moment generating functions and then taking the inverse Laplace transformation. However, this may be a tedious and complex process.

Alternatively, you can use numerical methods such as Monte Carlo simulation or bootstrapping to estimate the CDF. This approach may be simpler and more feasible, especially if you have a large number of populations.

As for the assumption of independence, if your data are truly independent, then the problem may simplify as the joint moment generating function of independent random variables is the product of their individual moment generating functions. However, it is important to assess the independence of your data before making this assumption.

I hope this helps and please let me know if you have any further questions or concerns. Good luck with your analysis!