(統計)Normal distribution，常態分配

 

Normal distribution，常態分配

$X \sim N(\mu,\sigma^2)$

$f(x)=\cfrac{1}{\sqrt{2\pi\sigma^2}}e^{\cfrac{(X-\mu)^2}{2\sigma^2}}$    ，$-\infty<x<\infty$

$M(t)=e^{\mu t+\frac{1}{2}\sigma^2t^2}$

$E(X)=\mu$

$Var(X)=\sigma^2$

Standard normal distribution，標準常態分配

$Z \sim N(0,1)$

$f(z)=\cfrac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}}$    ，$-\infty<z<\infty$

$Z=\cfrac{X-\mu}{\sigma}$

$M(t)=e^{\frac{1}{2}t^2}$

$E(Z)=0$

$Var(Z)=1$

常態分配加法性

$\displaystyle\sum_{i=0}^{n}a_iX_i \ \sim N(\displaystyle\sum_{i=0}^{n}a_i\mu_i,\displaystyle\sum_{i=0}^{n}a_i^2\sigma_i^2)$

二元常態分配

$(X_1,X_2)\sim N_2(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho_{12}) $

$f_{X_1,X_2}(x_1,x_2)=\cfrac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho_{12}^2}}e^{\displaystyle\cfrac{1}{2(1-\rho_{12}^2)}[(\cfrac{x_1-\mu_1}{\sigma_1})^2 -  2 \rho_{12} \cfrac{x_1-\mu_1}{\sigma_1}\cfrac{x_2-\mu_2}{\sigma_2}+(\cfrac{x_2-\mu_2}{\sigma_2})^2]}$


$,-\infty<x_1,x_2<\infty , \ -\infty<\mu_1,\mu_2<\infty,\ \ 0<\sigma_1,\sigma_2(\sigma_1,\sigma_2\neq 0),\ \ -1<\rho_{12}<1$

m.g.f

$M_{X_1X_2}(t_1,t_2)=e^ {\displaystyle  \mu_1 t_1+ \cfrac{\sigma_1^2t_1^2}{2}+ \mu_2 t_2+ \cfrac{\sigma_2^2t_2^2}{2}+\rho\sigma_1\sigma_2t_1t_2}$