Gaussian Distribution

Probability density function

\(X \sim \mathcal{N}(\mu, \sigma^2)\)
\begin{align*} p(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] \end{align*}

Q-function
- Definition
  \begin{align*} Q(x) \triangleq \int_x^{\infty} \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right)dx \end{align*}
- Properties
  \begin{align*} Q(-x) + Q(x) &= 1 \\ Q(0) &= \frac{1}{2} \end{align*}
Error function and complementary error function
- Definitions
  \begin{align*} \text{erf}(x) &\triangleq \frac{2}{\sqrt{\pi}} \int_0^x \exp\left(-x^2\right)dx \\ \text{erfc}(x) &\triangleq \frac{2}{\sqrt{\pi}} \int_x^{\infty} \exp\left(-x^2\right)dx \end{align*}
- Properties
  \begin{align*} \text{erf}(x) + \text{erfc}(x) &= 1 \\ \text{erf}(x) &= 1 - 2Q(\sqrt{2}x) \\ \text{erfc}(x) &= 2Q(\sqrt{2}x) \end{align*}