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Conditional Binomial

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  • Conditional Binomial

    I broke out my old Mathematical Statistics and Applications 7th Edition (On the SOA Exam P Sylllabus), Hopefully the people reading this have a copy of the book.

    I don't need the answer but I just want to make sure I'm on the right track with the problem because I feel there is no way to do it without seeing some nifty integration trick

    5.154 Supplemental Problems for Chapter 5

    it says p, (the proportion of patients who respond favorably to a drug) has a probability density function

    f(p) = 12p^2(1-p) for 0 <= p <= 1
    0 otherwise

    Suppose we have n patients, let random variable Y denote the number showing a favorable response.

    Find the unconditional probability distribution of Y for general n.

    My work:
    Clearly, Y|p has a binomial distribution.
    We know the distribution of p, so just apply the formula
    to get f(P,Y)..
    Great. But in doing so You get a nasty expression...good luck trying
    to integrate that to find the marginal of Y.

    Is this the right thinking or am I totally off. Any help appreicated. Thanks.