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Probability density for related variables

  1. Jan 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Say I calculated a probability density of a system containing m spins up (N is the total number of particles). The probabilities of being up and down are equal so this is easy to calculate. Let's call it ##\omega_m##. Then we define magnetization as ##M=2m-N## and it asks me to calculate the probability density of M.

    2. Relevant equations
    ##\omega_m(m)=\frac{1}{2^N}\frac{N!}{m!(N-m)!}##

    3. The attempt at a solution
    I'm not sure how to interpret a probability density of M. M can have a value between -N and N, so the probability, for example, of M=-N is the same as m=0. This suggests me that I can simply replace m for (M+N)/2 in its probability density expression. I don't know if that makes sense, the mathematical properties of these probability densities are no where to be found (at least not with this particular detail).
     
  2. jcsd
  3. Jan 28, 2015 #2
    I think my comprehensioon of that probability density improved a bit. It's the probability of the system to be in the state defined by N and m (or M), so my solution obtained from simply rewriting ##\omega## as ##\omega(N,M)## is right. Right?
     
  4. Jan 29, 2015 #3

    haruspex

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    Yes, except consider parity.
     
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