Definition: Normal distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Normal distribution ▷ Definition

Definition: Let $X$ be a random variable. Then, $X$ is said to be normally distributed with mean $\mu$ and variance $\sigma^2$ (or, standard deviation $\sigma$)

$$\label{eq:norm} X \sim \mathcal{N}(\mu, \sigma^2) \; ,$$

if and only if its probability density function is given by

$$\label{eq:norm-pdf} \mathcal{N}(x; \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right]$$

where $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.

 

Sources:

• Wikipedia (2020): "Normal distribution"; in: Wikipedia, the free encyclopedia, retrieved on 2020-01-27; URL: https://en.wikipedia.org/wiki/Normal_distribution.

Metadata: ID: D4 | shortcut: norm | author: JoramSoch | date: 2020-01-27, 14:15.