normal distribution

A normal distribution is a probability distribution which models the expected value of a large number of samples from a discrete random variable with finite mean and variance (see central limit theorem). A random variable which is normally distributed is denoted $X\sim \mathcal{N}(\mu ,{\sigma }^2)$, where $\mu$ is the mean and ${\sigma }^2$ is the variance.

Formally,
the standard normal distribution of a random variable $Z\sim \mathcal{N}(0,1)$ is given by the probability density function

and the general normal distribution of a random variable $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ is given by the probability density function

Visually,