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We are interested in the joint density of two Poisson distributions conditioned on their sum:

$$ P(X_1=n_1,X_2=n_2|X_1+X_2=n) $$

Where: $$ n_1 + n_2 = n $$

I am looking for a citation supporting the fact that this is a binomial. I was able to find a fairly straightforward blog post here https://blog.jpolak.org/?p=1924. Specifically, I'm looking for a citation that:

$$ \frac{P(X_1=n_1,X_2=n-n_1)}{P(X_1+X_2=n)} $$

is equivalent to:

$$ \binom{n}{n_1}p^{n_1}(1-p)^{n-n_1} $$

I know this seems fairly straightforward, but it would be nice to have a better citation than just a blog post.

kjetil b halvorsen
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a_crowell
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    Could you tell us why you cannot derive the result from first principles? – Xi'an Feb 24 '21 at 16:38
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    [Here is a generalization](https://stats.stackexchange.com/questions/84098/conditional-distribution-of-poisson-variables-given-sum-x-i) or even better https://stats.stackexchange.com/questions/429564/conditional-probability-poisson-and-exponential – kjetil b halvorsen Feb 24 '21 at 16:55
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    Kjetil, thank you for the generalization, that is definitely helpful! Xi'an the reason I am looking for a citation is that this is for a response to a reviewer who doesn't seem to have a strong grounding in probability. The paper is in an applied field and the reviewer seems to believe these are not equivalent. We could provide them with the derivation but I'm not sure how much good it would do (Plus it's always good to give credit where it is due with a citation if possible :) ). – a_crowell Feb 24 '21 at 17:13
  • @kjetilbhalvorsen The second link is very helpful as well - it provides a really intuitive way of getting at the problem without any calculation which is sort of perfect for our situation. Thanks! – a_crowell Feb 24 '21 at 17:28

1 Answers1

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If you still need a formal reference the following should do: W Feller, An Introduction to Probability Theory and its Applications, volume 1, third edition, chapter 9, section 9: Problems for solution, problem 6. That should also satisfy the reviewer it is not a very difficult problem!

kjetil b halvorsen
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  • Thanks you so much! I really appreciate you taking the time to help - the reference and explanation in the second link together should be more than enough. – a_crowell Feb 24 '21 at 17:35