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I was wondering if any classical distributions can be arrived at via a hypergeometric r.v. where the number of draws from the urn is also a random variable? Ideally I would like the sample space for the number of draws to be over [0,$\infty$], so a gamma distribution is my first thought.

Through some research I have found out and understood the proof for why a poisson r.v. with gamma distributed mean is a negative binomial r.v. So at worst I think I could use the poisson to approximate the hypergeometric, and thus the negative binomial result should work to approximate the hypergeometric with a gamma random number of draws (unless I am thinking about this completely wrong). However an analytic result would be preferable to an approximation.

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