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Find the maximum value of a summation

  1. Jul 18, 2011 #1
    Hi,
    What is the maximum value of the given summation in terms of [itex]k, l[/itex] and [itex]N[/itex] ?
    [tex]max_{0\leq x \leq k} \sum_{(l_1,l_2)\in A} \frac{N!}{(N-l_1-l_2)!l_1!l_2!} x^{l_1}(k-x)^{l_2}(1-k)^{N-l_1-l_2}[/tex]
    where [itex]A=\{(l_1,l_2)|l_1,l_2 \in \{0,1,2,...,N\} [/itex] and [itex]l_1+2l_2=l\}[/itex] and [itex]0<k<1[/itex].
    Thanks a lot for your participation.
     
    Last edited: Jul 18, 2011
  2. jcsd
  3. Jul 19, 2011 #2

    hunt_mat

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    Homework Helper

    Forgetting about all the constants for a moment, you have a function [itex]x^{a}(k-x)^{b}[/itex], the way to look at this is differentiate it ans et it to zero, do this for the simple function and see what you get.
     
  4. Jul 19, 2011 #3
    I had done it before. it is [itex](\frac{k l_1}{l_1+l_2})^{l_1} (\frac{k l_2}{l_1+l_2})^{l_2}[/itex].but the problem is that each term of the summation is maximized in different values of the given interval as [itex]l_1, l_2[/itex] vary.
     
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