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14:32
I will illustrate. Your question has two obvious keywords, so I first tried a search for questions with some upvotes that mention them both: stats.stackexchange.com/…. This had hundreds of hits and the top hit titles had little obvious relevance.
To refine the search, I noticed many hits referred to the negative binomial distribution. I therefore looked for posts that did not include the word "negative." A little risky, perhaps, because that's such a common word, but the objective is merely to find some relevant posts, not all of them.
Even that produced hundreds of hits. After a little reflection, it occurred to me that one key aspect of your question concerns the Poisson rate and focuses on the distribution (rather than estimation, etc.), so I looked for questions that mention both these terms: stats.stackexchange.com/…
This yielded just 20 results and the third one in the list has the title Q: Distribution of successes of poisson process followed by a binomial distribution. Le voila -- there's the duplicate.
Total time spent searching and thinking was under one minute. This is a typical experience and, typically, in my experience a minute of such iterated searching yields an answer in the majority of cases.
Perhaps that's because, through familiarity with this site, I have a good sense of which questions ought to have duplicates.
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