<p>Starting from the probability theory of continuous random variables and the central limit theorem, a rigorous mathematical proof is presented to show that the one-dimensional velocity components of particles in gas phase at thermal equilibrium can only be normally distributed if the physical properties are independent of direction. As such direction-independence is true for all gases, no matter whether they are ideal or not, a general distribution can be introduced. It is also shown that the particle speeds, which are the Euclidean norms of the velocity vectors, are always described by a chi distribution with three degrees of freedom, which converts into the Maxwell-Boltzmann speed distribution if the ideal gas law is valid. Furthermore, many of the formulas derived for ideal gases have analogs for real gases, which can be constructed by replacing <i>RT</i> (gas constant multiplied by temperature) terms by <i>pV</i>_m (pressure multiplied by molar volume).</p>