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A statistical system (whether classical or quantum mechanical) can be described in terms of the number of particles with a given energy, n(ε).  Restart.  In general, this function is itself the product of two functions: n(ε) = g(ε)f(ε).  g(ε) counts the number of states of a given energy and is also called the density of states.  f(ε) is the probable distribution of the particles through the states and is also called the occupancy. The form of these functions varies depending on the system of interest: whether governed by classical statistics or quantum statistics.

The animation shows you these distribution functions for a system of fermions (since these are fermions, the correct statistics is Fermi-Dirac) at a particular temperature. You can evaluate the integral of each function from any starting point and end point. For example, in order to find the total number of particles, you must integrate n(ε) over the entire energy range. How many particles are in this system?  Over what energy range are states filled (a probability of 1)?  Over what range are they empty (a probability of 0)?  For this system, is the degeneracy constant as a function of energy?