*Ecological Archives* M078-019-A5

**C. Wirth, J. W. Lichstein, J. Dushoff, A. Chen, and F. S. Chapin, III. 2008. White spruce meets black spruce: dispersal, postfire establishment, and growth in a warming climate. ***Ecological Monographs * 78:489–505.

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 .

LITERATURE CITED

Casella, G., and R. L. Berger. 2002. Statistical Inference. Second Edition. Duxbury Press. 688 pp.

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