Stat mech: Fermi-Dirac distribution

[davon806]

Homework Statement

Show that the FD distribution can be viewed as giving the probability that a given state ( of the prescribed
energy) is occupied.

Homework Equations

The Attempt at a Solution

Solution to this problem:

I understand the solution,but I took a different approach which gave a different answer.

For a quantum state i,denote Z_i as its partition function.Then for a single state distribution(2nd red box) :

For a fermion,n_i = 0 or 1. I want to find P(1) for the state i,so by the 2nd box it is P(1) = e^β(μ-ε_i) / Z_i , which is not the same as the FD distribution?

 

[DrClaude]

For a fermion,n_i = 0 or 1. I want to find P(1) for the state i,so by the 2nd box it is P(1) = e^β(μ-ε_i) / Z_i , which is not the same as the FD distribution?

What happens if you multiply the numerator and the denominator by $e^{-\beta (\mu - \epsilon)}$?