Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics — Henri Poincare
Couldn’t help writing the quote from this article (subscription required).
Why Gaussianity? by Kim, K. and Shevlyakov, G. (2008) IEEE Signal Processing Magazine, Vol. 25(2), pp. 102-113
It’s been a while since my post, signal processing and bootstrap from IEEE signal processing magazine, described as tutorial style papers on signal processing research and applications. Because of its tutorial style, the magazine delivers most up to date information and applications to people in various disciplines (their citation rate is quite high among scientific fields where data are collected via digitization except astronomy. This statement is solely based on my experience and no proper test was carried out to test this hypothesis). This provoking title, perhaps, will drag attentions about advances in signal processing from astronomers in future.
A historical account on Gaussian distribution, which goes by normal distribution among statisticians is given: de Moivre, before Laplace, found the distribution; Laplace, before Gauss, derived the properties of this distribution. The paper illustrates the derivations by Gauss, Herschel (yes, astronomer), Maxwell (no need to mention his important contribution), and Landon along with these following properties:
- the convolution of two Gaussian functions is another Gaussian function
- the Fourier transform of a Gaussian function is another Gaussian function
- the CLT
- maximizing entropy
- minimizing Fisher information
You will find pros and cons about Gaussianity in the concluding remark.