5.5 The normal distribution (continuous)

Definition 5.3 (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 \(\sigma^2>0\).