The Maxwell Boltzmann speed distribution, P(v), is given by:
P(v) = 4 pi (m/(2 pi k T))^(3/2) Exp[-m v^2/(2 k T)] v^2
P(v)dv is the probability that a molecule of mass m has a speed between v and v + dv. Let's denote the energy distribution by Q(E). Then Q(E)dE is the probability that a molecule has an energy between E and E + dE. To find Q(E), eliminate v in P(v)dv ...
The formula of the Maxwell Boltzmann translational energy distribution is derived.