Normal distribution
Normal distribution may refer to function
\(\displaystyle \varphi(x)=\exp(-x^2/2)/\sqrt{2\pi} \)
aka «Gauss bell» or «Gaussian exponential»; it is BellFunction.
Various notations appear in publications; it may cause confusions.
Normal distribution is good model to start with, if the researcher has no idea about possible distribution of quantities analyzed. Then, revealing of statistically-significant deviations may give a hint to more specific model.
Some properties of the «Gaussian exponential» are collected below.
This article is far from to be complete; use wikipedia or Citizendium or cited literature.
Integration
\(\displaystyle \int_{0}^{\infty} x^{2n} e^{-p x^2} \mathrm d x = \frac{(2n-1)!!}{2(2p)^n} \sqrt{\frac{\pi}{p}} = \frac{(2n-1)!!}{2^{n+1}p^{n+1/2}} \sqrt{\pi} \)
\(\displaystyle
\int_{0}^{\infty}
x^{2n+1} e^{-p x^2} \mathrm d x = \frac{n!}{2p^{n+1}}
\)
Case \( p=1/2\)
\(\displaystyle \varphi(x)=\exp(-x^2/2)/\sqrt{2\pi} \)
\(\displaystyle
\int_{0}^{\infty}
x^{2n} e^{- x^2/2}\ \mathrm d x = \frac{(2n-1)!!}{2} \sqrt{2\pi}
\)
\(\displaystyle \int_{-\infty}^{\infty} \varphi(x) \ \mathrm d x = 1 \)
\(\displaystyle \int_{-\infty}^{\infty} \varphi(x)\ x^2 \ \mathrm d x = 1 \)
Warning
References
http://fisica.ciens.ucv.ve/~svincenz/TISPISGIMR.pdf
Gradstyein, Ryzhik. Tables..
https://en.wikipedia.org/wiki/Normal_distribution
Keywords
«Elementary function», «Cosh», «Exp»,
