# Normal distribution probability density function for dummies

Could someone explain in plain English the parts of Gaussian PDF? Why $$\pi$$, why Euler constant etc.

$$f(x; \mu, \sigma^2) = \dfrac{1}{\sigma \sqrt{2\pi}} \exp\left\{-\frac{1}{2} \left( \frac{x - \mu}{\sigma}\right)^2\right\}.$$

• Uhh, I think a textbook might be better for this. Dec 20, 2019 at 18:40
• Check this answer. Dec 20, 2019 at 18:49
• That answer says everything except why the $x^2$ appears in the exponent. For that, see this. Dec 20, 2019 at 19:06
The simplest cumulant-generating function of a non-constant variable is quadratic, say $$i\mu t-\frac12\sigma^2t^2$$ (see here for further motivation). It can be shown the resulting distribution has mean $$\mu$$ and variance $$\sigma^2$$, and the PDF you cited. Because of how distributions respond to linear transformations, we need only check the $$\mu=0,\,\sigma=1$$ case, i.e. prove$$\varphi(t):=\exp-\frac12t^2\implies\int_{\Bbb R}\frac{1}{2\pi}\varphi(t)\exp(-itx)dt=\frac{1}{\sqrt{2\pi}}\exp-\frac{x^2}{2}dx.$$(This integral is the PDF, by the inversion formula.) The proportionality constants boil down to the $$\alpha=\frac12$$ special case of$$\int_{\Bbb R}\exp(-\alpha y^2)dy=\sqrt{\frac{\pi}{\alpha}}.$$Again, verification need only check $$\alpha=1$$. This has many proofs, the first here being the standard one in textbooks.