Derivation of the Average Velocity (Mean) from
Maxwell's Velocity Distribution Equation.
The average velocity from Maxwell's velocity distribution is obtained by adding up all of the velocities and dividing by the total number of molecules.

The AvgVel can also be obtained as the sum of each individual velocity times the probability for that velocity.
Because there are a very large number of molecules, it is OK to integrate to obtain the sum.
<= Shows n times the probability
<= Shows n3.
In order to integrate the above conveniently with MathCad, I will replace ¥ with a suitably large number. In this case, I will use 50000.
<= This is no longer the velocity distribution, but rather the velocity distribution multiplied by n, thus it has n3 rather than n2. In the next equation, I have let Mathcad integrate.
The first term, has a large negative exponent and can be considered to be zero, thus there is an immediated simplification to
<= this is the average velocity (mean) and I will simplify it.
For the purpose of showing that the algebra is correct, I will plug in some values.
Thus the arithmetic mean (average) velocity is