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Explicit variable from equation

  1. Oct 13, 2009 #1
    hi all!!

    I'm facing this problem and I can't get a solution, I want to explicit variable m from the following equation, I tried with binomial theorem to break up binomial coefficient but it was useless...
    can someone help me please?

    [tex]\sum_{i=n}^{n+m}\binom{n+m} {i}p^i(1-p)^{n+m-i} = k[/tex]

    all variables are known except m, and k is a known constant.
    virtually I'd like something in the form:
    [tex]m = g(n,p,k)[/tex]

  2. jcsd
  3. Oct 13, 2009 #2


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    Science Advisor

    Look at a couple of simple examples. Suppose m= n= 1. Then that says
    [tex]\sum_{i=1}^{2}\binom{2}{i}p^i(1-p)^{2-i} = k[/tex]
    which is
    [tex]\binom{2}{1}p(1- p)+ \binom{2}{2}p^2= 2p- 2p^2+ p^2= 2p- p^2= k[/tex]

    In general, that is an m+n degree polynomial in p and you are not going to find any explicit formula for the solution of such a polynomial.
  4. Oct 13, 2009 #3
    mm let me understand better...you are talking about a polynomial in p, but I told that p is known! my incognita is m... did I miss something?
  5. Oct 14, 2009 #4


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    Science Advisor

    Sorry, for some reason I thought you had said "p". But because the m occurs both in the binomial coefficient and in the exponent, I would say that there is much less likely to be an explicit formula for m.
  6. Oct 14, 2009 #5
    mm ok! :)
    do you see another way to calculate m (or an approximated value)??
    It's ok for me to get the minimum value of m that satisfies:

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