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Wei_Dai comments on Open Thread, August 2010-- part 2 - Less Wrong
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Here's my solution. The descendants of each initial blob spawn independently of descendants of other initial blobs, so this is a sum of N independent distributions. The number of descendants of one initial blob is obviously the geometric distribution. Googling "sum of independent geometric distributions" gives Negative binomial distribution as the answer.
Agreed - there are never more than N breeding blobs, each success increases P by one, and each failure reduces the remaining number of breeding blobs by one. Essentially, if r = N, X = P-N.
Thanks for answering several questions at once. :)
I don't think that's right. I don't have the math to show why yet, but my current working hunch says to make explicit your assumptions about whether the initial number of blobs, and the number of generations, are continuous or discrete, because the geometric distribution may not actually be right.