As I understood it, the 'normal distribution' is $$\frac{1}{\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2{\sigma}^2}\right)$$ Now according to this the 'normal probability density function' is $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2{\sigma}^2}\right)$$ and according to this at the top of page 466 (or 436 if you own the book) the 'normalised Gaussian distribution' is $$f(t)=\frac{1}{\tau \sqrt{2\pi}}\exp\left(\frac{-t^2}{2{\tau}^2}\right)$$
In the cases above, $\mu =$ mean, $\sigma =$ standard deviation and $\tau =\Delta t$
I am very confused; each time I search the web for answers I get different words and formulae, this post is designed to try to dispel this confusion.
Could someone please explain the meaning and difference between the formulae given above and the similarity and/or differences between the phrases used in the title and body?
Thank you.