  In this paper, the expression of the relativistic Maxwell-J\"uttner velocity
distribution was deduced in a detailed and didactic way. This distribution was
compared with the well-known Maxwell-Boltzmann distribution, which does not
take into account relativistic effects. Applications and comparisons between
the two distributions were carried out in three situations: i) protons at solar
surface temperature, ii) electrons at Lawson's criterion temperature, and iii)
protons at quasar temperatures. It was found that the value of the ratio
between the rest mass of the particle considered and the temperature of the
system determines the need to consider or not a relativistic treatment for the
velocity distribution function of the system.
