Derivation of the nonrelativistic Maxwellian speed distribution:

Let $v \equiv \vert \vec{V} \vert$ be the speed of a particle (e.g., an electron) of mass $m$ in a gas of temperature $T$. From thermodynamics, recall that the average kinetic energy is $k T / 2$ per degree of freedom (e.g., per spatial coordinate for a single particle), so
$$ { m \langle V_{\rm x}^2 \rangle \over 2} = { m \langle V_{\rm y}^2 \rangle \over 2} = { m \langle V_{\rm z}^2 \rangle \over 2} = { k T \over 2}$$
$$\langle V^2 \rangle = \langle V_x^2 \rangle + \langle V_y^2 \rangle + \langle V_z^2 \rangle = { 3 k T \over m}$$

Collisions eventually bring the gas into LTE, leading to identical Gaussian distributions for $V_{\rm x}$, $V_{\rm y}$, and $V_{\rm z}$. Writing out only the $x$-coordinate distribution $P(V_{\rm x})$ yields
$$P(V_{\rm x}) = {1 \over \sqrt{2 \pi} \sigma_{\rm x} } \exp \biggl( - {V_{\rm x}^2 \over 2 \sigma_{\rm x}^2} \biggr) ,$$
where $\sigma_{\rm x}$ is the rms (root mean square) value of $V_{\rm x}$. The definition of this rms is
$$\sigma_{\rm x}^2 \equiv \langle V_{\rm x}^2 \rangle = \int_{-\infty}^\infty V_{\rm x}^2 P(V_{\rm x})  d V_{\rm x} = \int_{-\infty}^\infty { V_{\rm x}^2 \over \sqrt{ 2 \pi} \sigma_{\rm x} } \exp \biggl( - {V_{\rm x}^2 \over 2 \sigma_{\rm x}^2} \biggr) d V_{\rm x}$$
$$\sigma_{\rm x}^2 = {1 \over \sqrt { 2 \pi} \sigma_{\rm x} } {1 \over 2} \sqrt{\pi} \biggl({1 \over 2 \sigma_{\rm x}^2} \biggr)^{-3/2} = {kT \over m} $$
so
$$ P(V_{\rm x}) = {1 \over \sqrt{2 \pi} } \biggl( {m \over k T } \biggr)^{1/2} \exp \biggl( - {m V_{\rm x}^2 \over 2 k T} \biggr) .$$

In three dimensions, by isotropy,
$$P(V_{\rm x}, V_{\rm y}, V_{\rm z}) d V_{\rm x}  d V_{\rm y}  d V_{\rm z} = P(V_{\rm x}) P(V_{\rm y}) P(V_{\rm z})  d V_{\rm x}  d V_{\rm y}  d V_{\rm z}$$
$$P(V_{\rm x}, V_{\rm y}, V_{\rm z}) = \biggl( { m \over 2 \pi k T} \biggr)^{3/2} \exp \biggl( - {m V^2 \over 2 k T} \biggr)$$
All velocities in the spherical shell of radius $V = (V_{\rm x}^2 + V_{\rm y}^2 + V_{\rm z}^2)^{1/2}$ correspond to the speed $v$, so
$$f(v) = 4 \pi V^2 P(V_{\rm x}, V_{\rm y}, V_{\rm z})$$
$$f(v) = {4 v^2 \over \sqrt{\pi}} \biggl( { m \over 2 k T} \biggr)^{3/2} \exp \biggl( - {m v^2 \over 2 k T} \biggr)$$

This is the desired result: the nonrelativistic Maxwellian distribution $f$ of speeds $v$ for particles of mass $m$ at temperature $T$.  If we normalize the speeds by the rms speed $(3 k T / m)^{1/2}$, the Maxwellian speed distribution looks like this: