What is Normal distribution?

$X$ follows the standard normal distribution $\mathcal N(0, 1)$ if its density on $\R$ is

(the bell curve). More generally, $X \sim \mathcal N(\mu, \sigma^2)$ if $\dfrac{X - \mu}{\sigma} \sim \mathcal N(0,1)$; then $\E(X) = \mu$ and $\V(X) = \sigma^2$.