# 对数正态分布

參數 μ=0 概率密度函數 μ=0 累積分佈函數 $\sigma \ge 0$ $-\infty \le \mu \le \infty$ $x \in [0; +\infty)\!$ $\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{\left[\ln(x)-\mu\right]^2}{2\sigma^2}\right)$ $\frac{1}{2}+\frac{1}{2} \mathrm{erf}\left[\frac{\ln(x)-\mu}{\sigma\sqrt{2}}\right]$ $e^{\mu+\sigma^2/2}$ $e^{\mu}$ $e^{\mu-\sigma^2}$ $(e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}$ $(e^{\sigma^2}\!\!+2)\sqrt{e^{\sigma^2}\!\!-1}$ $e^{4\sigma^2}\!\!+2e^{3\sigma^2}\!\!+3e^{2\sigma^2}\!\!-6$ $\frac{1}{2}+\frac{1}{2}\ln(2\pi\sigma^2) + \mu$ (参见原始动差文本) $\sum_{n=0}^{\infty}\frac{(it)^n}{n!}e^{n\mu+n^2\sigma^2/2}$is asymptotically divergent but sufficient for numerical purposes

$f(x;\mu,\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}} e^{-(\ln x - \mu)^2/2\sigma^2}$

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

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

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

## 与几何平均值和几何标准差的关系

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

## 矩

$\mu_1=e^{\mu+\sigma^2/2}$
$\mu_2=e^{2\mu+4\sigma^2/2}$
$\mu_3=e^{3\mu+9\sigma^2/2}$
$\mu_4=e^{4\mu+16\sigma^2/2}$

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

## 局部期望

$g(k)=\int_k^\infty (x-k) f(x)\, dx$

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

## 参数的最大似然估计

$f_L (x;\mu, \sigma) = \frac 1 x \, f_N (\ln x; \mu, \sigma)$

$\begin{matrix} \ell_L (\mu,\sigma | x_1, x_2, ..., x_n) & = & - \sum _k \ln x_k + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n) = \\ \\ \ & = & \operatorname {constant} + \ell_N (\mu, \sigma | \ln x_1, \ln x_2, \dots, \ln x_n). \end{matrix}$

$\widehat \mu = \frac {\sum_k \ln x_k} n, \ \widehat \sigma^2 = \frac {\sum_k {\left( \ln x_k - \widehat \mu \right)^2}} n.$

## 相关分布

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