22 Gaussian Distribution
A random variable X \in \mathbb{R} follows the Gaussian distribution
X \sim \mathcal{N} (\mu, \sigma^{2}) \quad \mu \in \mathbb{R}, \sigma^{2} \in \mathbb{R},
if X follows the standard bell curve.
\mathbb{P}_{X} (x) = \frac{ 1 }{ \sigma \sqrt{2 \pi} } \exp \left[ - \frac{ (x - \mu)^{2} }{ 2 \sigma^{2} } \right]
\mathbb{E}_{X} [X] = \mu
\mathrm{Var} (X) = \sigma^{2}
Corollary 22.1 (Linear transformation) Let X \sim \mathcal{N} (\mu, \sigma^{2}). For any a, b \in \mathbb{R} with a \neq 0,
a X + b \sim \mathcal{N} (a \mu + b, a^{2} \sigma^{2}).
Standard normal distribution
Corollary 22.2 (Standardization) Let X \sim \mathcal{N} (\mu, \sigma^{2}). Taking a = \frac{1}{\sigma}, b = -\frac{\mu}{\sigma} in Corollary 22.1 gives the standardization of X
\frac{ X - \mu }{ \sigma } \sim \mathcal{N} (0, 1).
Definition 22.1 (Standard normal distribution) Applying the standardization to a Gaussian random variable X gives a new random variable that follows the standard normal distribution Z
Z \sim \mathcal{N} (0, 1).
Its CDF is usually denoted by \Phi
\Phi (z) = \mathbb{P} (Z \leq z), \quad z \in \mathbb{R},
which does not have a closed form.
Definition 22.2 (Standard normal quantile) For \alpha \in (0, 1), the upper \alpha quantile and the lower \alpha quantile of the standard normal distribution are the values that leave probability \alpha in the right and left tail respectively
\mathbb{P} (Z \geq z_{\alpha}) = \alpha, \qquad \mathbb{P} (Z \leq -z_{\alpha}) = \alpha, \qquad Z \sim \mathcal{N} (0, 1),
where z_{\alpha} \in \mathbb{R} is the upper \alpha quantile and -z_{\alpha} \in \mathbb{R} is the lower \alpha quantile.
Corollary 22.3 (Two-sided quantile) For \alpha \in (0, 1),
\mathbb{P} \left( -z_{\alpha / 2} \leq Z \leq z_{\alpha / 2} \right) = 1 - \alpha, \quad Z \sim \mathcal{N} (0, 1).