I was just trying to calculate the electrical conductivity for a Fermi-Dirac distribution and a Maxwell-Boltzmann distribution, and I ended up with the same result:
$$\sigma=\frac{ne^{2}\tau}{m}$$
My question is, how come this is? Ending up with the same result even though it is different distributions?
The model you're looking at is called the Drude-Sommerfeld model. Sommerfeld looked at the motion of free electrons taking into account Pauli exclusion and found exactly the same coincidence that you did.
The reason is that you're assuming a parabolic dispersion relation, and all the particles get shifted in momentum space by a certain amount ''dp'', which has the same value in both Drude model and Drude-Sommerfeld model. This means that in some sense they are all drifting with a certain speed (remember you assumed parabolic dispersion), and that speed is the same in both instances.
Of course you can also see conduction as being a Fermi surface phenomenon. In this case, you have much fewer particles participating in conduction, but they are all travelling much faster (Fermi speed is huge, compared to ideal gas rms speed). The two factors (less electrons, but more speed) cancel out.
Another way to put it, notice that neither answer depends on temperature. If you increase the temperature of the Drude-Sommerfeld model, then eventually you will get an ideal gas where Pauli exclusion is irrelevant. The formulas have to agree, in that case.
Both MB and Drude assume a dilute gas; the collision mean free time is large. On the other hand one wouldn't expect a large effect from the Pauli exclusion principle until the gas was relatively dense.