Normal Distribution

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

• $\mu \in R,{\sigma }^2>0$
• $f_X(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2},x\in \mathbb{R}$
• $F_X(x)={\int }_{-\infty }^x{f}_X(t)dt$
• $M_X(t)=e^{\mu t+\frac{1}{2}{\sigma }^2{t}^2},t\in \mathbb{R}$
• $\mathbb{E}[X]=\mu$
• $\text{var}(X)={\sigma }^2$
  For the standard normal distribution $Z=\frac{X-\mu }{\sigma }\sim \mathcal{N}(0,1)$
• $f_Z(z)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{z^2}{2}}$
• $F_Z(z)=\Phi (z)={\int }_{-\infty }^z{f}_Z(t)dt$
• $M_Z(t)=e^{\frac{1}{2}{t}^2}$
• $\mathbb{E}[Z]=0$
• $\text{var}(Z)={\sigma }_Z=1$
• $\Phi (-x)=1-\Phi (x)$
  Table for $\varphi (x)$