# Probability density for related variables

## Homework Statement

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.

## Homework Equations

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

## 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).

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?

haruspex
Homework Helper
Gold Member
This suggests me that I can simply replace m for (M+N)/2 in its probability density expression.
Yes, except consider parity.