# Random Velocity

Considering gas in Maxwellian velocity distribution, the distribution function for the velocity is as follows:

 (E.1)

This gives the one-dimensional random velocity as
 (E.2)

If we observe emissions from such a gas, the emission line is broaden due to the Doppler shift. Using equation (E.2), the HWHM (half width of half maximum: the line width measured from the the center of the emission line to the point of the half intensity; see Fig.E.1) of the emission line is
 (E.3)