Maxwell–Boltzmann distribution

Key facts

Parameters

0"}}'> a > 0 {\displaystyle a>0}

Support

x ∈ ( 0 ; ∞ ) {\displaystyle x\in (0;\infty )}

Pdf

2 π x 2 a 3 exp ⁡ ( − x 2 2 a 2 ) {\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {x^{2}}{a^{3}}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)} (where exp is the exponential function )

Cdf

erf ⁡ ( x 2 a ) − 2 π x a exp ⁡ ( − x 2 2 a 2 ) {\displaystyle \operatorname {erf} \left({\frac {x}{{\sqrt {2}}a}}\right)-{\sqrt {\frac {2}{\pi }}}\,{\frac {x}{a}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)} (where erf is the error function )

Mean

μ = 2 a 2 π {\displaystyle \mu =2a{\sqrt {\frac {2}{\pi }}}}

Mode

2 a {\displaystyle {\sqrt {2}}a}

Variance

σ 2 = a 2 ( 3 π − 8 ) π {\displaystyle \sigma ^{2}={\frac {a^{2}(3\pi -8)}{\pi }}}

Skewness

γ 1 = 2 2 ( 16 − 5 π ) ( 3 π − 8 ) 3 / 2 ≈ 0.48569 {\displaystyle \gamma _{1}={\frac {2{\sqrt {2}}(16-5\pi )}{(3\pi -8)^{3/2}}}\approx 0.48569}

Excess kurtosis

γ 2 = 4 ( − 96 + 40 π − 3 π 2 ) ( 3 π − 8 ) 2 ≈ 0.10816 {\displaystyle \gamma _{2}={\frac {4(-96+40\pi -3\pi ^{2})}{(3\pi -8)^{2}}}\approx 0.10816}

Entropy

ln ⁡ ( a 2 π ) + γ − 1 2 {\displaystyle \ln \left(a{\sqrt {2\pi }}\right)+\gamma -{\frac {1}{2}}}

via Wikipedia infobox