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Homework Help: Show that the maximum of a multinomial distribution is given by N1=N2= =Ns=N/s

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data

    So one of the problems in my statistical mechanics textbook (McQuarrie problem 1-50) asks to show that the maximum of a multinomial distribution is given for N1=N2=...=Ns=N/s

    In the book, he shows that for large N, the binomial coefficients when taken as a function are maximized at N/2. Then he expands the natural log around that point and shows that it takes on the form of a gaussian function.

    The derivation is found at this link

    and the exact wording of the problem is here. Problem 50.

    The first thing I don't really get where he pulls the N1*=N/2 from. I know that to actually show that isn't too difficult, but I can't just assume the same result for the multinomial case because that's what I'm trying to show.

    2. Relevant equations

    The multinomial coefficients are given by
    [tex]\frac{N!}{\prod_{j=1}^r N_j!}[/tex] where [itex]N_1+N_2+\cdots+N_r=N[/itex]

    3. The attempt at a solution

    Just from working with the coefficients expression subject to the above constraint results in the following when maximized with Lagrange multipliers
    [tex]N\ln N - \sum N_j\ln N_j - \lambda(\sum N_j -N)=0=F(N_j)[/tex]
    [tex](-\sum_{j=1}^s\ln N_j-\lambda)\, dN_j=0[/tex] which is the total differential
    If I write out the partial derivatives with respect to each term I get
    [tex]\frac{\partial F}{\partial N_1}=-\ln N_1-1 - \lambda=0[/tex]

    [tex]\frac{\partial F}{\partial N_2}=-\ln N_2-1 - \lambda=0[/tex]

    [tex]\frac{\partial F}{\partial N_3}=-\ln N_3-1 - \lambda=0[/tex]

    etc

    So i conclude that [itex]N_1=N_2=N_3=\cdots=N_s[/itex]. At the same time, if [itex]sN_j=N[/itex] implies that [itex]N_j=N/s[/itex]
     
    Last edited: Mar 23, 2010
  2. jcsd
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