Graduate Is the binomial a special case of the beta binomial?

[Ad VanderVen]

TL;DR

On Wikipedia one can read in the article Beta-binomial distribution:

It also approximates the binomial distribution arbitrarily well for large $\alpha$ and $\beta$.

What is the meaning of 'arbitrarily'?

On Wikipedia one can read in the article Beta-binomial distribution:

> It also approximates the binomial distribution arbitrarily well for
> large $\alpha$ and $\beta$.

where 'It' refers to the beta-binomial distribution. What does 'arbitrarily well' mean here?

 

[fresh_42]

See
https://en.wikipedia.org/wiki/Convergence_of_random_variables
and the links therein.

 

[pbuk]

What does 'arbitrarily well' mean here?

It means the same here as it always means: you can get as close to the target as you want by aiming carefully.

In this case, you can get a distribution as close as you like to the binomial if you choose large enough $\alpha$ and $\beta$.

Is the binomial a special case of the beta binomial?​

No (because there are no finite values of $\alpha, \beta$ that make them equal), but it is the limiting case as $\alpha$ and $\beta$ tend to infinity.