Consider an ideal gas obeying the Maxwell-Boltzmann distribution i.e

$$f(v) = \bigg(\frac{m}{2 \pi k\_{B} T}\bigg)^{3/2} \exp \left(-\frac{m v^{2}}{2 k\_{B} T} \right) \, .$$

The probability distribution in 3D velocity space ($v^{2} = v\_{x}^2 + v\_{y}^2 + v\_{z}^2$). How might you determine the average speed the particles are moving at, $\langle |v\_{z}| \rangle$, in one direction?

Additionally if my ideal gas is now confined to a hemisphere in velocity space i.e we have the conditions
$- \infty \leq v\_{x}, v\_{y} \leq \infty$ and $ 0 \leq v\_{z} \leq \infty$
but it still has a Maxwell Boltzmann velocity distribution (except I think the normalization factor on $f(v)$ should change) then what is the average speed, or velocity, in the z direction $\langle v\_{z} \rangle$, will this be the same as $|\langle v\_{z}| \rangle$ from the previous answer?