I'm dealing with some biological data (read counts from gene expression measurements). For each individual we have 2 measurements, which correspond to two different version of the same gene. These measurements are discrete count data, and generally we would model them with with a negative binomial distribution, since they contain plenty of noise in a addition to the poisson distributed shot noise.
What I"m interested in is the ratio between counts. I'm given to understand that if we consider two poisson distributed random variables, and condition on their total counts, then the count for one in particular will be binomial distributed (correct me if I'm wrong). My question is this - what if we replace the poisson distributions here with negative binomial distributions? What is the distribution of the counts for each one, conditional on the total? Is it a beta binomial?