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StatMech: Binomial approximation

  1. Sep 22, 2013 #1
    1. The problem statement, all variables and given/known data

    The question requires me to approximate binomial distribution to get poisson distribution.
    Show that N!/(N-n)!=N^n.

    2. Relevant equations

    N!/n!(N-n)! p^n q^(N-n)=Binomial distribution



    3. The attempt at a solution

    I expanded N!/(N-n)! and got: (N-1)(N-2)(N-3)....(N-n+2)(N-n+1). This didn't help me in getting the required approximation. So, then I wrote it as follows:( N-(n-(n-1)) ) ( N-( n- (n-2) ) )...( N-(n-2) ) ( N-(n-1) ).
    It seem to have further complicated the question.
    A little help please.:redface:
    Thank you.
     
  2. jcsd
  3. Sep 22, 2013 #2

    UltrafastPED

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    Science Advisor
    Gold Member

    Look at how the Stirling formula is derived ... it should be quite helpful.
     
  4. Sep 22, 2013 #3
    I looked at the stirling formula derivation but I don't know how it is helpful here.
    So I have solved it the other way.
    [N-(n-(n-0))] [N-(n-(n-1))] [N-(n-(n-2))] [N-(n-(n-3))]......[N-( n-(3) )][N-( n-(2) )][N-( n-(1) )]
    For N>>n, using this approximation once, I get n terms:
    [N-n] [N-n] [N-n] ...... [N-n] [N-n] [N-n]
    using the approximation again,
    [N] [N] [N] ......[N] [N] [N] =N^n

    My question is can I use this approximation selectively like I did in the above two steps.
    Secondly, how was sterling formula derivation helpful? I used the x! formula and I get exponential(-n). Because n<<N, this term is big, making the entire answer zero.
     
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