${\displaystyle \text{d}N=A\cdot \text{exp}\left[\frac{-m\left(u^2+v^2+w^2\right)}{2kT}\right]\text{d}u\cdot \text{d}v\cdot \text{d}w}$

where

${\displaystyle A=N{\left[\frac{m}{\left(2\pi \cdot kT\right)}\right]}^{3/2}}$

N is the total number of particles
m is the mass of a particle
T is the thermodynamic temperature
k is the Boltzmann constant

NOTE – dN/N represents the probability that a particle has its components of velocity within the intervals considered.