  We derive a Lorentz invariant distribution of velocities for a relativistic
gas. Our derivation is based on three pillars: the special theory of
relativity, the central limit theorem and the Lobachevskyian structure of the
velocity space of the theory. The rapidity variable plays a crucial role in our
results. For $v^2/c^2 \ll 1$ and $1/\beta=kT/2 m_0 c^2 \ll 1$ the distribution
tends to the Maxwell-Boltzmann distribution. The mean $\langle v^2 \rangle$
evaluated with the Lorentz invariant distribution is always smaller than the
Maxwell-Boltzmann mean and is bounded by $\langle v^2 \rangle/c^2=1$. This
implies that for a given $\langle v^2 \rangle$ the temperature is larger than
the temperature estimated using the Maxwell-Boltzmann distribution. For
temperatures of the order of $T \sim {10^{12}}~ K$ and $T \sim {10^{8}}~ K$ the
difference is of the order of $10 \%$, respectively for particles with the
hydrogen and the electron rest masses.
