Maxwell 分布律

Maxwell 分布律（麦克斯韦速度分布律）是统计物理学中理想气体分子模型下分子速度服从的一种概率分布。

模型

设非负连续型随机变量$X$的概率密度函数为 $$f(x) = \sqrt{\dfrac{2}{\pi}} \dfrac{x^2}{\sigma^3} \text{e}^{-\frac{x^2}{2\sigma^2}}, \quad x \geqslant 0.$$ 我们就说随机变量$X$服从参数为$\sigma$的 Maxwell 分布。

它的规范性体现在 $\begin{align} \int_0^{+\infty} f(x) \mathrm{d}x & = \int_0^{+\infty} \sqrt{\dfrac{2}{\pi}} \dfrac{x^2}{\sigma^3} \text{e}^{-\frac{x^2}{2\sigma^2}} \mathrm{d}x \\ & = \sqrt{\dfrac{2}{\pi}} \int_0^{+\infty} \dfrac{2 \sigma^2 t}{\sigma^3} \text{e}^{-t} \dfrac{\sigma}{\sqrt{2t}} \mathrm{d}t \\ & = \dfrac{1}{\sqrt{\pi}} \int_0^{+\infty} t^\frac{1}{2} \text{e}^{-t} \mathrm{d}t \\ & = \dfrac{1}{\sqrt{\pi}} \Gamma(\dfrac{3}{2}) = 1. \end{align}$ 上述做代换$t = \dfrac{x^2}{2\sigma^2}, x = \sigma\sqrt{2t}.$

数字特征

数学期望和方差分别是$E(X) = \dfrac{2\sqrt{2}}{\sqrt{\pi}}\sigma, D(X) = \left( 3 - \dfrac{8}{\pi} \right) \sigma^2.$证明如下 $$\begin{align} E(X) & = \int_0^{+\infty} \sqrt{\dfrac{2}{\pi}} \dfrac{x^3}{\sigma^3} \text{e}^{-\frac{x^2}{2\sigma^2}} \mathrm{d}x \\ & = \sqrt{\dfrac{2}{\pi}} \int_0^{+\infty} \dfrac{2 \sigma^3 t\sqrt{2t}}{\sigma^3} \text{e}^{-t} \dfrac{\sigma}{\sqrt{2t}} \mathrm{d}t \\ & = \dfrac{2\sqrt{2}\sigma}{\sqrt{\pi}} \int_0^{+\infty} t \text{e}^{-t} \mathrm{d}t = \dfrac{2\sqrt{2}\sigma}{\sqrt{\pi}} \Gamma(2) = \dfrac{2\sqrt{2}}{\sqrt{\pi}}\sigma. \\ D(X) & = E(X^2) - (E(X))^2 \\ & = \int_0^{+\infty} \sqrt{\dfrac{2}{\pi}} \dfrac{x^4}{\sigma^3} \text{e}^{-\frac{x^2}{2\sigma^2}} \mathrm{d}x \\ & = \sqrt{\dfrac{2}{\pi}} \int_0^{+\infty} \dfrac{4 \sigma^4 t^2}{\sigma^3} \text{e}^{-t} \dfrac{\sigma}{\sqrt{2t}} \mathrm{d}t - \dfrac{8}{\pi} \sigma^2 \\ & = \dfrac{4\sigma^2}{\sqrt{\pi}} \int_0^{+\infty} t^\frac{3}{2} \text{e}^{-t} \mathrm{d}t - \dfrac{8}{\pi} \sigma^2 \\ & = \dfrac{4\sigma^2}{\sqrt{\pi}} \Gamma(\dfrac{5}{2}) - \dfrac{8}{\pi} \sigma^2 \\ & = 3\sigma^2 - \dfrac{8}{\pi} \sigma^2 = \left( 3 - \dfrac{8}{\pi} \right) \sigma^2. \\ \end{align}$$

参考资料

1. 李贤平, 《概率论基础（第3版）》, 高等教育出版社, 北京, 2010-04, ISBN 978-7-0402-8890-2.