The probability that an electron state at energy level E is occupied is
given by the electron occupancy distribution function, 
, and for holes by the hole occupancy distribution function, 
. The basic result is as follows:
where 
.
We first show that these two results are consistent with each other. This requires:
since each state is either occupied or empty. Actually, this can be
considered to be a statement of the Pauli exclusion principle, which
prevents more than one electron from occupying the same state. To show this,
we first define 
, observe that 
, and then perform a direct
substitution:
The appearance of the constant kT in the above formulas requires a study of statistical thermodynamics that is beyond the scope of these notes. Instead, we derive the general form of these functions, called the Fermi-Dirac statistic:
