Open Thread, August 2010-- part 2

NancyLebovitz

This thread is for the discussion of Less Wrong topics that have not appeared in recent posts. If a discussion gets unwieldy, celebrate by turning it into a top-level post.

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.

13Wei Dai16y

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.

0Pavitra16y

Offhand, I think you would also need to know the number of generations. I'll have to do some pen-and-paper work to work out what the distribution looks like.

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