Is the binomial a special case of the beta binomial?

In summary, the beta-binomial distribution approximates the binomial distribution very well for large values of alpha and beta, meaning that it can get as close as desired to the binomial by choosing appropriate values for alpha and beta. The binomial is not a special case of the beta-binomial, but it becomes the same as the beta-binomial as alpha and beta approach infinity.
  • #1
Ad VanderVen
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TL;DR Summary
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?
 
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  • #3
Ad VanderVen said:
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##.

Ad VanderVen said:

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.
 

FAQ: Is the binomial a special case of the beta binomial?

What is a binomial distribution?

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (usually referred to as success and failure).

What is a beta binomial distribution?

A beta binomial distribution is a probability distribution that combines the binomial distribution with the beta distribution. It is used to model the number of successes in a fixed number of independent trials, where the probability of success in each trial follows a beta distribution.

How is the binomial distribution related to the beta binomial distribution?

The binomial distribution is a special case of the beta binomial distribution, where the beta distribution parameters are set to 1. This means that the probability of success in each trial is fixed and does not vary.

What are the advantages of using the beta binomial distribution over the binomial distribution?

The beta binomial distribution allows for more flexibility in modeling the probability of success in each trial, as the beta distribution parameters can be adjusted to fit different scenarios. It also accounts for overdispersion, which is common in real-world data and cannot be captured by the binomial distribution.

In what situations would the beta binomial distribution be a better fit than the binomial distribution?

The beta binomial distribution would be a better fit when the probability of success in each trial is not fixed and can vary. It is also more appropriate when dealing with data that shows overdispersion, such as in count data or biological data.

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