Probability density for related variables

1. Jan 28, 2015

diegzumillo

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. Jan 28, 2015

diegzumillo

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?

3. Jan 29, 2015

haruspex

Yes, except consider parity.