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.
Best Answer
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.