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Soa 43

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  • Soa 43

    On this problem how might we use the binomial distribution? I know the solution is using the NB but cant you use (x/n)(ncx)(p^x)(q^n-x) somehow?

  • #2
    Originally posted by claytonmccandless View Post
    On this problem how might we use the binomial distribution? I know the solution is using the NB but cant you use (x/n)(ncx)(p^x)(q^n-x) somehow?
    Isn't this "(x/n)(ncx)(p^x)(q^n-x)" the negative binomial distribution? If that is what you want to use, then you are using the NB are you not?

    I thought that

    fb = (nCx)(p^x)(q^(n-x))
    and
    fnb = (x/n)(nCx)(p^x)(q^(n-x))

    so fnb = (x/n)fnb

    So I would think that the answer to your question is that yes, you can use a binomial distribution, but in order to use it in this case, you have to multiply it be the coefficient (x/n). And by doing so, you actually end up turning it into the negative binomial distribution.

    I believe that you can think of the negative binomial as just a more restrictive case of the binomial distribution. For the Binomial distribution, x is the random variable and the PDF gives the probability of x successes in a fixed number of n trials. For the Neg Binomial distribution, n is the random variable and the PDF gives the probability that n trials are required for a fixed number of x successes to occur. So in the B.D. we just count the # of successes in n trials, but in the N.B.D. we count the number of trials it takes to get to x successes. So even if you use the same values for x and n in the B.D. and the N.B.D. you get different probabilities because the B.D. counts all the chances of having x success in n trials (the xth success can be before the last trial) but the N.B.D. only counts the chances of having the x successes where the last success occurs on the last trial. So you have x-1 successes in the first n-1 trials, and the xth success occurs on the nth trial. That is a subset of the chances of any occurrence of x successes in n trials. So that is why the N.B.D. is a fraction of the B.D.

    I actually show further detail on how to use the N.B.D. for this question in another thread here:
    http://www.actuary.com/actuarial-dis...ad.php?p=65709

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