Standard multivariate normal distribution

Let $Z_i \overset{iid}{\sim} N(0,1) \space (i=1,\cdots,n)$
$Z=(Z_1,\cdots,Z_n)^T$ has multivariate normal distribution with mean vector:0, covariance matrix: $I_n$

• pdf
  $f_Z(z)=\prod_{i=1}^{n} f_{Z_i}(z_i)=(2 \pi)^{-n/2} exp(-\frac{1}{2} \sum_{i=1}^{n} z_i^2)$
  
  
• mgf
  For any $t=(t_1,\cdots,t_n)^T \in \mathbb{R}^n$,
  $M_Z(t)=E[exp(t^T z)]=E[\prod_{i} exp(t_i z_i)]=\prod_i E[exp(t_i z_i)]$(b/c $Z_i's$ are indep.)$=\prod_i exp(\frac{1}{2} t_i^2)=exp(\frac{1}{2}t^T t)$

Multivariate normal distribution

Let $X=\Sigma^{1/2}Z+\mu$, then $E(X)=E(\Sigma^{1/2}Z+\mu)=\mu$ and $Var(X)=\Sigma^{1/2} Var(Z)\Sigma^{1/2}=\Sigma$

• pdf
  $f(x)=det(2 \pi \Sigma)^{-1/2} exp(-\frac{1}{2}(x-\mu)^T \Sigma^{-1} (x-\mu)), \space x \in \mathbb{R}^n$
  $\because$
  
  
• mgf
  $M_X(t)=E[exp(t^T X)]=exp(t^T \mu) E[exp(t^T \Sigma^{1/2}Z)]=exp(t^T \mu) M_Z(\Sigma^{1/2}t)=exp(t^T \mu)exp(\frac{1}{2}t^T \Sigma^{1/2} \Sigma^{1/2}t)=exp(t^T \mu + \frac{1}{2}t^T \Sigma t), \space t \in \mathbb{R}^n$

✔︎ 동치명제

$X \sim N_n(\mu,\Sigma), \space \Sigma \in \mathbb{R}^{n \times n}$: nonnegative definite
<=> $X \overset{d}{\equiv} \Sigma^{1/2}Z + \mu$ with $Z=(Z_1, \cdots, Z_n)^T$ and $Z_i \overset{iid}{\sim} N(0,1)$
<=> $M_X(t)=exp(\mu^T t+\frac{1}{2}t^T \Sigma t)$ for any $t \in \mathbb{R}^n$
<=> $a^TX \sim N(a^T \mu,a^T \Sigma a)$ for any $a \in \mathbb{R}^n$