For a classical ideal gas in thermal equilibrium, the three Cartesian velocity components are distributed independently according to Gaussian factors because the kinetic energy is $$E=\frac12m(v_x^2+v_y^2+v_z^2).$$ The velocity probability density is therefore $$f(\mathbf v)=\left(\frac{m}{2\pi k_BT}\right)^{3/2} \exp\left(-\frac{mv^2}{2k_BT}\right).$$

Converting from velocity-space volume $d^3v$ to speed shells gives the Maxwell speed distribution $$P(v)=4\pi v^2\left(\frac{m}{2\pi k_BT}\right)^{3/2} \exp\left(-\frac{mv^2}{2k_BT}\right),\qquad v\ge0.$$ The factor $v^2$ reflects the increasing number of velocity directions available at larger radius in velocity space, while the exponential suppresses very large kinetic energies.

The most probable speed is $$v_{\rm mp}=\sqrt{\frac{2k_BT}{m}},$$ the mean speed is $$\langle v\rangle=\sqrt{\frac{8k_BT}{\pi m}},$$ and the root-mean-square speed is $$v_{\rm rms}=\sqrt{\frac{3k_BT}{m}}.$$ These are different because the distribution is skewed.

Increasing temperature broadens the distribution and shifts it to higher speeds. Increasing particle mass at fixed temperature shifts it lower. The Maxwell-Boltzmann distribution connects microscopic molecular velocities to macroscopic temperature and underlies kinetic explanations of pressure, transport and collision rates.