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How does one model the Binomial distribution where the probability of success is the result of another Binomial distribution.

For example, say I make 10 coin tosses many times and record the number of heads (H). Then for each set (i) of 10 coin tosses I put Hi black marbles, and 10-Hi white marbles in a jar and make 50 draws with replacement. How would I model the distribution of the black marble draws taking into account their dependence on the previous Binomial distribution that generated their probability of success.

kjetil b halvorsen
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user20629
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2 Answers2

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This is known as a compound distribution. Some compound distributions simplify, and can be recognized as another well-known distribution, but I don't think this binomial-binomial compound is one of those. (There is another type of binomial-binomial compound which does simplify to just a binomial, where you toss all coins, and then reflip the heads.)

I think the simplest way to handle the distribution is as a mixture of $11$ different binomial distributions parametrized by the number of heads in the initial batch, from $0$ to $10$. There are other possibilities based on recognizing this as a compound of a multinomial distribution.

Douglas Zare
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Thanks D. Zare for the response. I'm the OP. I spent a little more time looking into this and I believe another solution is to use a beta-binomial distribution (http://en.wikipedia.org/wiki/Beta-binomial_distribution). The beta-binomial requires alpha and beta shape parameters like a beta distribution. In the context of the situation described above, alpha = beta = 5, which corresponds to the size of the first "coin toss" binomial. For small values of alpha and beta the beta-binomial distribution is more dispersed than a binomial. As alpha and beta get larger the distribution converges on the binomial.

user20629
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