Gaussian Distribution

probability-theory

Definition

Gaussian Distribution

The Gaussian distribution is a continuous distribution that is parameterised by $\mu \in \mathbb{R}$ (mean) and ${\sigma }^2\in {\mathbb{R}}_{>0}$ (variance).

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

Properties

Affine Transform

Let $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ with cdf $F_X$ and $a,b\in \mathbb{R}$.

$$Y=a\cdot X+b\sim \mathcal{N}(a\mu +b,a^2{\sigma }^2)$$

Standardisation

Let $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ with cdf $F_X$ and $a,b\in \mathbb{R}$.

Affine transformation to standard location and scale:

$$Z=\frac{X-\mu }{\sigma }\sim \mathcal{N}(0,1)$$

This works for any random variable $X$ for which the first two moments exist. That is $Z=\frac{X-\mathbb{E}[X]}{\sqrt{\mathbb{V}ar(X)}}$ has:

• Expectation: $\mathbb{E}[Z]=0$
• Variance: $\mathbb{V}ar(Z)=1$.

Quantiles

Let $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ with cdf $F_X$ and $a,b\in \mathbb{R}$.

The $p$-quantile $x_p$ of $X$ is related to the standard normal $p$-quantile $z_p$:

$$x_p=\mu +\sigma z_p$$

where $z_p={\Phi }^{-1}(p)$ for $p\in (0,1)$ and $x_p=F_X^{-1}(p)$.

Symmetries

For the standard normal density, cdf, and $p$-quantiles holds:

• $f_Z(z)=f_Z(-z)$
• $\Phi (-z)=1-\Phi (z)$
• $z_p=-z_{1-p}$

Standard

Definition

Standard Gaussian Distribution

The standard Gaussian distribution is a Gaussian distribution with a mean $\mu =0$ and a variance ${\sigma }^2=1$.

Link to original

Probability Density Function

The probability density function is given by:

$$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\cdot \exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$$

Parameter effects: