# Distribution of the ratio between two Negative binomial distributions

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?

• Do the two negative binomial distributions have the same parameter values? Commented Sep 26, 2016 at 13:22
• No, I don't have any reason to think they do, although they're likely to be quite similiar. Commented Sep 26, 2016 at 15:01
• Can these counts be 0? Commented Sep 27, 2016 at 4:43
• The total must be at least 1, and will probably get thresholded higher than that at something like 20 or 50. Commented Sep 27, 2016 at 15:30
• ttu-ir.tdl.org/ttu-ir/bitstream/handle/2346/59954/… Commented Oct 2, 2016 at 19:07