Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 34D / iii / Solution

The Boltzmann distribution factors between momentum and position. Normalizing the height factor givesIn the tall-box limit this becomes $\boxed{\beta mg{e}^{-\beta mgz}dz}$ on $z\ge 0$, an exponential distribution with scale height $k_{B}T/(mg)$.