P.Mean: Mixtrure of two binomial distributions (created 2012-12-31).
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The binomial distribution might itself be the mixture of two or more binomial distributions. It should be obvious that the mixture of two binomial distributions is also binomial, but here are the details. You start with n patients and there is a probability pi1 that the patient survives the first round of exclusions and a conditonal probability pi2 that the patient survives the second round of exclusions assuming that they survived the first round. Let i be the number surviving the first round and j be the number surviving the second round. The probability of observing j is

To get a total of j successes across two binomials, you need to get i successes out of n for the first binomial (i>=j) and then j successes out of i for the second binomial. Simplify as before to get

Notice that we put (n-j)! in the numerator inside the sum and in the denominator outside the sum. This will make it easier to simplify things down the road. Substitute k=i-j and m=n-j to get

The terms inside the summation look like the traditional binomial probabilities, but the two probabilities pi1(1-pi2) and (1-pi1) are not quite complementary. With a bit of work, you can show that these bionomial-like probabilities inside the summatio simplify to something a bit less than 1.0, yielding the following formula.

This is a binomial distribution with probability of success equal to the product of the two probabilities of success, pi1 and pi2. This can be easily extended to a mixture of k binomial distributions.

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