4.6 Gaussian Random Variables

• It’s also called the normal random variable.

• This is the most important continuous random variable, due to its wise use among all scientific disciplines.

• Definition: Let $X$ be a Gaussian random variable with parameters $\mu, \sigma^2$, then:

  • $f_X(x) = \dfrac{1}{\sqrt{2\pi\sigma^2}}\text{ exp }\left\{ -\dfrac{(x-\mu)^2}{2\sigma^2} \right\}$
  • Notation: $X \sim Gaussian(\mu, \sigma^2) \sim N(\mu, \sigma^2)$

• Theorem: If $X \sim Gaussian(\mu, \sigma^2)$, then:

  • $\mathbb{E}[X] = \mu$.
  • $Var[X] = \sigma^2$.