5.5 The normal distribution (continuous)

Definition 5.5 (Normal distribution) The normal distribution with mean $\mu$ and variance $\sigma^2$ has the PDF \[ f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2}\right\}\qquad\text{for $-\infty<x<\infty$} \] for $-\infty < \mu < -\infty$ and $\sigma^2 > 0$.