# 对数正态分布

参数 概率密度函數μ=0 累積分布函數μ=0 ${\displaystyle \sigma \geq 0}$${\displaystyle -\infty \leq \mu \leq \infty }$ ${\displaystyle x\in [0;+\infty )\!}$ ${\displaystyle {\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {\left[\ln(x)-\mu \right]^{2}}{2\sigma ^{2}}}\right)}$ ${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\mathrm {erf} \left[{\frac {\ln(x)-\mu }{\sigma {\sqrt {2}}}}\right]}$ ${\displaystyle e^{\mu +\sigma ^{2}/2}}$ ${\displaystyle e^{\mu }}$ ${\displaystyle e^{\mu -\sigma ^{2}}}$ ${\displaystyle (e^{\sigma ^{2}}\!\!-1)e^{2\mu +\sigma ^{2}}}$ ${\displaystyle (e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}}$ ${\displaystyle e^{4\sigma ^{2}}\!\!+2e^{3\sigma ^{2}}\!\!+3e^{2\sigma ^{2}}\!\!-6}$ ${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\ln(2\pi \sigma ^{2})+\mu }$ (参见原始动差文本) ${\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}$is asymptotically divergent but sufficient for numerical purposes
${\displaystyle f(x;\mu ,\sigma )={\frac {1}{x\sigma {\sqrt {2\pi }}}}e^{-(\ln x-\mu )^{2}/2\sigma ^{2}}}$

${\displaystyle \mathrm {E} (X)=e^{\mu +\sigma ^{2}/2}}$

${\displaystyle \mathrm {var} (X)=(e^{\sigma ^{2}}-1)e^{2\mu +\sigma ^{2}}.\,}$

${\displaystyle \mu =\ln(\mathrm {E} (X))-{\frac {1}{2}}\ln \left(1+{\frac {\mathrm {var} (X)}{\mathrm {E} (X)^{2}}}\right),}$
${\displaystyle \sigma ^{2}=\ln \left(1+{\frac {\mathrm {var} (X)}{\mathrm {E} (X)^{2}}}\right).}$

## 与几何平均值和几何标准差的关系

3σ 下界 ${\displaystyle \mu -3\sigma }$  ${\displaystyle \mu _{\mathrm {geo} }/\sigma _{\mathrm {geo} }^{3}}$
2σ 下界 ${\displaystyle \mu -2\sigma }$  ${\displaystyle \mu _{\mathrm {geo} }/\sigma _{\mathrm {geo} }^{2}}$
1σ 下界 ${\displaystyle \mu -\sigma }$  ${\displaystyle \mu _{\mathrm {geo} }/\sigma _{\mathrm {geo} }}$
1σ 上界 ${\displaystyle \mu +\sigma }$  ${\displaystyle \mu _{\mathrm {geo} }\sigma _{\mathrm {geo} }}$
2σ 上界 ${\displaystyle \mu +2\sigma }$  ${\displaystyle \mu _{\mathrm {geo} }\sigma _{\mathrm {geo} }^{2}}$
3σ 上界 ${\displaystyle \mu +3\sigma }$  ${\displaystyle \mu _{\mathrm {geo} }\sigma _{\mathrm {geo} }^{3}}$

## 矩

${\displaystyle \mu _{1}=e^{\mu +\sigma ^{2}/2}}$
${\displaystyle \mu _{2}=e^{2\mu +4\sigma ^{2}/2}}$
${\displaystyle \mu _{3}=e^{3\mu +9\sigma ^{2}/2}}$
${\displaystyle \mu _{4}=e^{4\mu +16\sigma ^{2}/2}}$

${\displaystyle \mu _{k}=e^{k\mu +k^{2}\sigma ^{2}/2}.}$

## 局部期望

${\displaystyle g(k)=\int _{k}^{\infty }(x-k)f(x)\,dx}$

${\displaystyle g(k)=\exp(\mu +\sigma ^{2}/2)\Phi \left({\frac {-\ln(k)+\mu +\sigma ^{2}}{\sigma }}\right)-k\Phi \left({\frac {-\ln(k)+\mu }{\sigma }}\right)}$

## 参数的最大似然估计

${\displaystyle f_{L}(x;\mu ,\sigma )={\frac {1}{x}}\,f_{N}(\ln x;\mu ,\sigma )}$

${\displaystyle {\begin{matrix}\ell _{L}(\mu ,\sigma |x_{1},x_{2},...,x_{n})&=&-\sum _{k}\ln x_{k}+\ell _{N}(\mu ,\sigma |\ln x_{1},\ln x_{2},\dots ,\ln x_{n})=\\\\\ &=&\operatorname {constant} +\ell _{N}(\mu ,\sigma |\ln x_{1},\ln x_{2},\dots ,\ln x_{n}).\end{matrix}}}$

${\displaystyle {\widehat {\mu }}={\frac {\sum _{k}\ln x_{k}}{n}},\ {\widehat {\sigma }}^{2}={\frac {\sum _{k}{\left(\ln x_{k}-{\widehat {\mu }}\right)^{2}}}{n}}.}$

## 相关分布

• 如果 ${\displaystyle Y=\ln(X)}$ ${\displaystyle X\sim \operatorname {Log-N} (\mu ,\sigma ^{2})}$ ，则 ${\displaystyle Y\sim N(\mu ,\sigma ^{2})}$ 正态分布
• 如果 ${\displaystyle X_{m}\sim \operatorname {Log-N} (\mu ,\sigma _{m}^{2}),\ m={\overline {1...n}}}$  是有同样 ${\displaystyle \mu }$  参数、而 ${\displaystyle \sigma }$  可能不同的统计独立对数正态分布变量 ，并且 ${\displaystyle Y=\prod _{m=1}^{n}X_{m}}$ ，则 ${\displaystyle Y}$  也是对数正态分布变量：${\displaystyle Y\sim \operatorname {Log-N} \left(n\mu ,\sum _{m=1}^{n}\sigma _{m}^{2}\right)}$

## 参考文献

• 对数正态分布, Aitchison, J. and Brown, J.A.C. (1957)
• Log-normal Distributions across the Sciences: Keys and Clues页面存档备份，存于互联网档案馆, E. Limpert, W. Stahel and M. Abbt,. BioScience, 51 (5), p. 341–352 (2001).
• 对数正态分布特性, John Hull, in Options, Futures, and Other Derivatives 6E (2005). ISBN 0-13-149908-4
• Eric W. Weisstein et al. 对数正态分布页面存档备份，存于互联网档案馆） at MathWorld. Electronic document, 2006年10月26日造訪.