# Maxwell–Boltzmann distribution

Parameters Probability density function Cumulative distribution function ${\displaystyle a>0}$ ${\displaystyle x\in (0;\infty )}$ ${\displaystyle {\sqrt {\frac {2}{\pi }}}{\frac {x^{2}e^{-x^{2}/\left(2a^{2}\right)}}{a^{3}}}}$ ${\displaystyle \operatorname {erf} \left({\frac {x}{{\sqrt {2}}a}}\right)-{\sqrt {\frac {2}{\pi }}}{\frac {xe^{-x^{2}/\left(2a^{2}\right)}}{a}}}$ where erf is the error function ${\displaystyle \mu =2a{\sqrt {\frac {2}{\pi }}}}$ ${\displaystyle {\sqrt {2}}a}$ ${\displaystyle \sigma ^{2}={\frac {a^{2}(3\pi -8)}{\pi }}}$ ${\displaystyle \gamma _{1}={\frac {2{\sqrt {2}}(16-5\pi )}{(3\pi -8)^{3/2}}}}$ ${\displaystyle \gamma _{2}=4{\frac {\left(-96+40\pi -3\pi ^{2}\right)}{(3\pi -8)^{2}}}}$ ${\displaystyle \ln \left(a{\sqrt {2\pi }}\right)+\gamma -{\frac {1}{2}}}$