# 拉普拉斯分布

参数 概率密度函數 累積分布函數 ${\displaystyle \mu \,}$ 位置参数（实数）${\displaystyle b>0\,}$ 尺度参数（实数） ${\displaystyle x\in (-\infty ;+\infty )\,}$ ${\displaystyle {\frac {1}{2\,b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,}$ 参见正文部分 ${\displaystyle \mu \,}$ ${\displaystyle \mu \,}$ ${\displaystyle \mu \,}$ ${\displaystyle 2\,b^{2}}$ ${\displaystyle 0\,}$ ${\displaystyle 3\,}$ ${\displaystyle 1+\ln(2\,b)}$ ${\displaystyle {\frac {\exp(\mu \,t)}{1-b^{2}\,t^{2}}}\,\!}$ for ${\displaystyle |t|<1/b\,}$ ${\displaystyle {\frac {\exp(\mu \,i\,t)}{1+b^{2}\,t^{2}}}\,\!}$

## 概率分布、概率密度以及分位数函数

${\displaystyle f(x|\mu ,b)={\frac {1}{2b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,\!}$
${\displaystyle ={\frac {1}{2b}}\left\{{\begin{matrix}\exp \left(-{\frac {\mu -x}{b}}\right)&{\mbox{if }}x<\mu \\[8pt]\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{matrix}}\right.}$

 ${\displaystyle F(x)\,}$ ${\displaystyle =\int _{-\infty }^{x}\!\!f(u)\,\mathrm {d} u}$ ${\displaystyle =\left\{{\begin{matrix}&{\frac {1}{2}}\exp \left(-{\frac {\mu -x}{b}}\right)&{\mbox{if }}x<\mu \\[8pt]1-\!\!\!\!&{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{matrix}}\right.}$ ${\displaystyle =0.5\,[1+\operatorname {sgn}(x-\mu )\,(1-\exp(-|x-\mu |/b))]}$

${\displaystyle F^{-1}(p)=\mu -b\,\operatorname {sgn}(p-0.5)\,\ln(1-2|p-0.5|)}$

## 生成拉普拉斯变量

${\displaystyle X=\mu -b\,\operatorname {sgn}(U)\,\ln(1-2|U|)}$

## 相关分布

• 如果 ${\displaystyle Y=|X-\mu |}$  并且 ${\displaystyle X\sim \mathrm {Laplace} }$ ，则 ${\displaystyle Y\sim \mathrm {Exponential} }$ 指数分布
• 如果 ${\displaystyle Y=X_{1}-X_{2}}$ ${\displaystyle X_{1},\,X_{2}\sim \mathrm {Exponential} }$ ，则 ${\displaystyle Y\sim \mathrm {Laplace} }$

## 统计推断

### 参数估计

${\displaystyle {\hat {b}}={\frac {1}{N}}\sum _{i=1}^{N}\left\vert x_{i}-{\hat {\mu }}\right\vert }$