분포의 특성

• pdf of $X$
  $f(x)=\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x- \mu)^2}{2 \sigma^2}} \space (-\infty<x<\infty,-\infty<\mu<\infty,\sigma^2>0)$
  
  $\int_{-\infty}^{\infty}f(x)dx=1?$
  
  
• 기댓값
  $E(X)=\mu$
  pf)
  
  
• 분산
  $Var(X)=\sigma^2$
  pf)
  
  
• mgf
  $M_X(t)=e^{\mu t+\frac{(\sigma t)^2}{2}} \space (-\infty<t<\infty)$
  pf)
  

관련 정리

Theorem)
$Z \sim N(0,1)$일 때, $Z^2 \sim \chi^2(1)$
pf)

✔︎ 주의사항
$N(0,1)$ 을 따르는 $Z$의 cdf는 닫힌 형태를 갇고 있지 않기 때문에, 표시할 때 편의를 위해 $\Phi(x)$로 표시한다.

$\Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2 \pi}}e^{-\frac{t^2}{2}}dt$

다변수 정규분포

• Joint pdf of $\underline{X}$

  Let $\underline{X}=(X_1,X_2,...,X_n)^T, \underline{\mu}=(E(X_1),E(X_2),...,E(X_n))^T, |\sum| = det(\sum)$
  
  $f(\underline{X}) = (2 \pi)^{-\frac{n}{2}} |\sum|^{-\frac{1}{2}} exp(-\frac{1}{2} (\underline{x} - \underline{\mu})^T \sum^{-1} (\underline{x} - \underline{\mu}) ) \\ \underline{X} \sim N_n(\underline{\mu}, \sum)$

• bivariate case

  $\begin{pmatrix} X_1\\ X_2 \end{pmatrix} \sim N_2(\begin{pmatrix} \mu_1\\ \mu_2 \end{pmatrix}, \begin{pmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2\\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{pmatrix})$
  
  $|\sum| = \begin{vmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \\ \end{vmatrix}=(1- \rho)^2 \sigma_1^2 \sigma_2^2$
  
  $\sum^{-1} = \frac{1}{(1- \rho)^2 \sigma_1^2 \sigma_2^2}\begin{pmatrix} \sigma_2^2 & -\rho \sigma_1 \sigma_2 \\ -\rho \sigma_1 \sigma_2 & \sigma_1^2 \\ \end{pmatrix}$
  
  $(\underline{x} - \underline{\mu})^T \sum^{-1} (\underline{x} - \underline{\mu}) = \frac{1}{(1- \rho)^2}[\frac{(x_1 - \mu_1)^2}{\sigma_1^2} -2 \rho \frac{(x_1 - \mu_1)(x_2 - \mu_2)}{\sigma_1 \sigma_2} + \frac{(x_2 - \mu_2)^2}{\sigma_2^2}]$
  
  $f(x_1,x_2) = (2 \pi)^{-1} ((1- \rho)^2 \sigma_1^2 \sigma_2^2)^{-1/2} exp(-\frac{1}{2} (\underline{x} - \underline{\mu})^T \sum^{-1} (\underline{x} - \underline{\mu})) = \\ \frac{1}{2 \pi \sqrt{(1- \rho)^2 \sigma_1^2 \sigma_2^2}} exp[-\frac{1}{2(1 - \rho)^2}(\frac{(x_1 - \mu_1)^2}{\sigma_1^2} -2 \rho \frac{(x_1 - \mu_1)(x_2 - \mu_2)}{\sigma_1 \sigma_2} + \frac{(x_2 - \mu_2)^2}{\sigma_2^2})]$
  
  If $\rho=0$,
  $f(x_1,x_2) = \frac{1}{2 \pi \sqrt{\sigma_1^2 \sigma_2^2}} exp[-\frac{1}{2} (\frac{(x_1 - \mu_1)^2}{\sigma_1^2}- \frac{(x_2 - \mu_2)^2}{\sigma_2^2})]=\frac{1}{\sqrt{2 \pi \sigma_1^2}} exp(-\frac{(x_1 - \mu_1)^2}{2 \sigma_1^2}) \frac{1}{\sqrt{2 \pi \sigma_2^2}} exp(-\frac{(x_2 - \mu_2)^2}{2 \sigma_2^2})$
  => $X_1,X_2$ are indep.
  => If $X_i$'s follows normal dist., $Cov(\underline{X})=0 <=>$ $X_i's$ are indep.