Consider an unknown number of new-born animal cubs,B, which can be split into two groups - those born from subadult animals,B1and those born from adult animals,B2, whereB = B1+B2. We sample these cubs, with each of the two groups having their own individual-level probability of being selected,p1andp2, respectively. The number of cubs sampled can be denoted asb1andb2, respectively.

I want to build a likelihood for the total number of cubs,B, given that I knowp1,p2,b1, andb2, to be used in a larger joint likelihood. If I were interested in building a likelihood for the two differentgroupsof cubs, I could write:

L(B1 | p1,b1) = [B1 choose b1] * (p1)^{b1} * (1 - p1)^{B1 - b1}

L(B2 | p2,b2) = [B2 choose b2] * (p2)^{b2} * (1 - p2)^{B2 - b2}

What I want, however, is

L(B | p1,b1,p2,b2) = ?

The full likelihood already has a large number of parameters to estimate, so I'd rather just have a single parameter for the number of new-born cubs each year, as opposed to having two parameters for each year - new-born cubs with sub-adult parents and new-born cubs with adult parents.

Any help would be greatly appreciated! A better-formatted version of the equations can be found in this thread: https://stats.stackexchange.com/ques...wo-likelihoods