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We have a pool of items, termed as item A, generated following a Poisson distribution. We use a pair of items A to produce an item B with success rate $r\in(0,1)$. My question is: is B Poisson distributed or not? If yes, how to prove it.

More generically, we can produce an item C by using a pair of item B, an item E by using an item A and an item B, etc, all with certain success rate, thus forming a probabilistic production tree with a root. My question is whether the root is Poisson distributed or not? If yes, how to compute its production rate?

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It is not. I am assuming that you have your poisson samples numbered 1,...,n as if they are the arrivals of a poisson process, and you try to form a new one out of 1 & 2, succeeding with probability r, then out of 3 & 4, then out of 5 & 6 etc. It is easy to write down the probability that you have k pairs to work with, because it is the probability that there are either 2k or 2k+1 in the poisson sample, and you can just look at it and see that it is not poisson. This does the case r=1, which you exclude, but if it were poisson for all r < 1, it would also be true in the limit of r=1. You can write down some sort of distribution for the resulting number of b's, but I don's think it is anything familiar.

Of course you know that if it were not for for the complication of taking pairs, you would have a 'selected poisson' and it is poisson distributed.

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  • $\begingroup$ Thank you for your answer. What kind of process is it then (e.g., is it close to Poisson so that I can approximate)? Could you pls elaborate slightly for me (e.g., I did not understand the second paragraph)? $\endgroup$
    – lchen
    Commented Oct 23, 2022 at 9:11

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