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:

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