Gaussian Distribution

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

• Definition

  $$\begin{array}{r}Q(x)\triangleq {\int }_x^{\mathrm{\infty }}\frac{1}{\sqrt{2\pi }}\exp (-\frac{x^2}{2})dx\end{array}$$

• Properties

  $$\begin{array}{rl}Q(-x)+Q(x)& =1\\ Q(0)& =\frac{1}{2}\end{array}$$

• Definitions

  $$\begin{array}{rl}\text{erf}(x)& \triangleq \frac{2}{\sqrt{\pi }}{\int }_0^x\exp (-x^2)dx\\ \text{erfc}(x)& \triangleq \frac{2}{\sqrt{\pi }}{\int }_x^{\mathrm{\infty }}\exp (-x^2)dx\end{array}$$

• Properties

  $$\begin{array}{rl}\text{erf}(x)+\text{erfc}(x)& =1\\ \text{erf}(x)& =1-2Q(\sqrt{2}x)\\ \text{erfc}(x)& =2Q(\sqrt{2}x)\end{array}$$