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I understand how to set up a binomial probability distribution. I'm trying to extend my understanding to the beta-binomial. On Wikipedia, there is a beta-binomial example given at https://en.wikipedia.org/wiki/Beta-binomial_distribution. I can follow the computations to get m1 and m2 and from there (6.23 and 42.31 respectively, n=12), hat-alpha and hat-beta (using the method of moments -- 34.1350 and 31.6085 respectively). I can reproduce in the example table the values for the row "Predicted (Binomial p = 0.519215)", but I can't figure out how to reproduce the values for the row "Predicted (Beta-Binomial)". How were those values computed?

Here's the data from the table (the number of male children among the first 12 children of family size 13 in 6115 families taken from hospital records in 19th century Saxony, so x ranges from 0-12):

Number of males:

0 1 2 3 4 5 6 7 8 9 10 11 12

Observed:

3 24 104 286 670 1033 1343 1112 829 478 181 45 7

Beta:

2.3 22.6 104.8 310.9 655.7 1036.2 1257.9 1182.1 853.6 461.9 177.9 43.8 5.2
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