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I've seen questions like this and this, but it hasn't quite answered my question. How intertwined are the Beta and Binomial distributions?

A quick sidenote: the Poisson distribution and the exponential distribution are two faces of the same coin. The Poisson distributions is a count process for events whereas the exponential distribution models time between those same events.

I know the Beta distribution is a convenient way to model the uncertainty in $p$ in a Binomial distribution because the Beta distribution is flexible (with the $\alpha$ and $\beta$ parameters) and exists between 0 and 1 (just like probabilities should be). But is there more to the story? Are the Beta distribution and Binomial distribution two faces of the same coin as well?

mnmn
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  • Have you searched our site yet? Try [beta binomial](https://stats.stackexchange.com/search?q=beta+binomial). A nice trick is to focus on the upvoted posts: [beta binomial score:2](https://stats.stackexchange.com/search?q=beta+binomial+score%3A2) narrows the field a bit and [this search](https://stats.stackexchange.com/search?q=beta+binomial+-%22beta-binomial%22+score%3A2) excludes the closely associated "beta-binomial" distribution. There's still a large amount of information already posted about this. An explicit account appears at https://stats.stackexchange.com/questions/4659. – whuber Aug 14 '19 at 13:16

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