隨機變數 $X\sim N(\mu,\sigma)$, 它的 pdf 如下

$\displaystyle f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\displaystyle-\frac{(x-\mu)^2}{2\sigma^2}} \mbox{ for } -\infty < x < \infty$

The normal distribution with a mean of $\mu$ and a standard deviation of $\sigma$ is denoted by $N(\mu, \sigma)$.

Interpreting the parameters, we can see in follow that a change of mean from $\mu_1$ to a larger value $\mu_2$ merely slides the bell-shaped curve along the axis until a new center is established at $\mu_2$. There is no change in the shape of the curve.

A different value for the standard deviation results in a different maximum height of the curve and changes the amount of the area in any fixed interval about $\mu$ as follow. The position of the center does not change if only $\sigma$ is changed.