Appendix E. Derivation of negative binomial distribution for the number of recruits.
We assume that seed rain (S) comes from a negative binomial distribution with parameters q (probability of success) and r (shape), so that the probability of s seeds arriving is
with mean and variance (Casella and Berger 2002). We assume that each seed has a probability p of establishing and surviving, so that the number of recruits (N) comes from a binomial distribution with s trials and probability of success p . The conditional probability of n seeds recruiting given seed rain s is then
The unconditional, or marginal, probability of n seeds recruiting is
where u ≡ s - n and α ≡ 1 - (1 - p)(1 - q). The summand is a negative binomial probability, so
which is negative binomial with mean and variance .
Casella, G., and R. L. Berger. 2002. Statistical Inference. Second Edition. Duxbury Press. 688 pp.