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Perplexed comments on Open Thread, August 2010-- part 2 - Less Wrong
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A quick probability math question.
Consider a population of blobs, initially comprising N individual blobs. Each individual blob independently has a probability p of reproducing, just once, spawning exactly one new blob. The next generation (an expected N*p individuals) has the same probability for each individual to spawn one new blob, and so on. Eventually the process will stop, with a total blob population of P.
The question is about the probability distribution for P, given N and p. Is this a well-known probability distribution? If so, which? Even if not, are there things that can be said about it which are mathematically obvious? (Not obvious to me, obviously. I'd be interested in which gaps in my math education I'm revealing by even asking these questions.)
After G generations, each blob has a probability q=p^G of having a descendant. So, it seems to me that P will be distributed as a binomial with q and N as parameters.
The blobs don't reproduce with probability p in any given generation, they reproduce with probability p ever. The scenario doesn't require generations in the sense you seem to be thinking of, it could all happen within 1 second, or a first generation blob might reproduce after the highest generation blob that reproduces has already done so.
Oh, ok. I thought the blobs died each generation. A shrinking population. Instead they go into nursing homes. A growing population which stabilizes once everyone is geriatric.
Got it. Wei pretty clearly has the solution. Negative Binomial distribution
Pretty damned obvious, actually, that (P-N) is distributed as a negative binomial where r is set to N; failure = failure to reproduce; success = birth.