Wolfram Alpha:

```Fermi-Dirac Statistics
-----------------------

The Fermi-Dirac distribution differs from the
classical Maxwell-Boltzmann distribution because
the particles it describes are indistinguishable.

Particles are considered to be indistinguishable
if their wave packets overlap significantly.  Two
particles can be considered to be distinguishable
if their separation is large compared to their
DeBroglie wavelength.

For example, the condition of distinguishability
is met by molecules in an ideal gas under ordinary
conditions.  On the other hand, two electrons in
the 1st shell of an atom are indistinguishable
because of the large overlap of their wavefunctions.

Let:

gi = the number of states with energy Ei
ni = the number of electrons in each energy state,
Ei

The total number of ways of distributing ni
electrons in gi states, Wi, with the restriction
that only one electron can occupy each state,
is given by:

- -
Wi = | g | =  gi!/(gi - ni)!ni!
| n |
- -

This the binomial coefficient!

The number of possible ways to fit the electrons
in the number of avalable states is called the
multiplicity function.

The multiplicity function for the whole system
is the product of the multiplicity functions
for each energy Ei:

W = ΠiWi = Πi[g!/(gi - ni)!ni!]

Simplify by taking the log of each side:

lnW = ΣilnWi

= Σiln[g!/(gi - ni)!ni!]

We can using Stirling's Approximation which says
that ln(n!) = nln(n) - n when n is large to get:

lnW = Σi[gilngi - nilnni - (gi - ni)ln(gi - ni)]

Now maximize lnW using the method of Lagrange
multipliers and the constraints Σini = N  and
E = ΣiniEi

Therefore,

lnW = Σi[gilngi - nilnni - (gi - ni)ln(gi - ni) - αni - βniEi]

Now the derivative of each of the i's must
equal 0.  Thus we can ignore the Σ.

∂lnW/∂ni = ln{(gi - ni)/ni} - α - βEi = 0

ln{(gi - ni)/ni} = α + βEi

(gi - ni)/ni = exp(α + βEi)

gi - ni = niexp(α + βEi)

gi = ni(1 + exp(α + βEi))

∴ ni = gi/(1 + exp(α + βEi))

The occupation probability = Pi = ni/gi

= 1/(1 + exp(α + βEi))

Let α = -EF/KBT and β = 1/KBT where EF is the
FERMI ENERGY.