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»,