Normal Distribution

Definition

$X\sim N(\mu ,{\sigma }^2)$ where $\mu \in \mathbb{R}$ is the location parameter(mean), and ${\sigma }^2\in {\mathbb{R}}_0^{+}$ is the scale parameter(variance)

Standard Normal Distribution

$X\sim N(0,1)$ $f(x)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{x^2}{2}\right)\in \mathbb{R}$

Properties

PDF

$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{1}{2}(\frac{x-\mu }{\sigma }{)}^2\right)\in \mathbb{R}$

Mean

$E(X)=\mu$

Variance

$\mathrm{Var}(X)={\sigma }^2$

MGF

$M_X(t)=\exp \left(\mu t+\frac{{\sigma }^2{t}^2}{2}\right)$

Higher Order Moments

$E(Z^k)=\{\begin{array}{l}E(Z^{2k})=\frac{(2k)!}{2^{k}k!}\\ E(Z^{2k+1})=0\end{array}$

$E(X^k)=\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{k}{j}\right){\sigma }^{j}E(Z^j){\mu }^{k-j}$

Sum of Normally Distributed Random Variables

Let $X_i\sim N({\mu }_i,{\sigma }_i^2)$ be independent random variables following normal distribution, then $\sum _{i=1}^n{a}_i{X}_i\sim N\left(\sum _{i=1}^n{a}_i{\mu }_i,{\sum }_{i=1}^n{a}_i^2{\sigma }_i^2\right)$

Relationship with Chi-squared Distribution

Let $Z\sim N(0,1)$ be a standard normal distribution, then $Z^2\sim {\chi }^2(1)$

Facts

Let $X\sim N(0,{\sigma }^2)$, then $\mathrm{Var}(X^2)=2{\sigma }^4$