Dear all. The Gaussian function is mostly used in statistics, physics and engineering. Some examples include:
1. Gaussian functions are the Green's function for the homogeneous and isotropic
2. The convolution of a function with a Gaussian is also known as a Weierstrass transform
3. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator
3. The atomic and molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals
4. Gaussian functions are also associated with the vacuum state in quantum field theory
5. Gaussian beams are used in optical systems, microwave systems and lasers
6. Gaussian functions are used to define some types activation function of artificial neural networks
7. Simple cell response in primary visual cortex has a Gaussian function modeled by a sine wave
8. Fourier transforms of Gaussian is a Gaussian
10. Easy and efficient approximation for signals analysis and fitting (Gaussian process, gaussian mixture model, kalman estimator , ...)
11. Discribe the Shape of the UV−Visible Absorption
12. Used in Time-frequency analysis (Gabor Transform)
13. Central Limit Theorem (CLT) : Sum of independent random variables tends toward a normal distribution
14. The Gaussian function serves well in molecular physics, where the number of particles is closed to the Avogadro number NA = 6.02214076×1023 mol−1 ( NA is defined as the number of particles that are contained in one mole)
15. ...
Why Gaussian is everywhere ?
https://chemistry.stackexchange.com/questions/33932/generating-neat-orbitals-surfaces-from-molden-wfn-files
https://en.wikipedia.org/wiki/Gaussian_function
https://en.wikipedia.org/wiki/Gaussian_beam
Lots of things are better described better by distributions based on, for example, Poisson functions, binomial functions and special cases of each of these. It is the Central Limit Theorem that you include that allows transformations or great amounts of data to be analyzed using a Gaussian function or approximations in the limit of other distributions.
I agree that in statistics, the Central Limit Theorem (CLT), when dealing with the distributions of means and totals, is a big reason for the gaussian distribution to have gotten the misleading alias "normal" distribution, as the use of the CLT is a big area of importance,* but you mention other cases which also deserve attention. I think that the wide use of this function has much to do with the usefulness of exponentials in nature, and
(x - mu)/sigma
relates range to standard deviation, something like an inverse relative standard deviation, such that we might use this to describe various natural phenomen. But there are various exponential and trigonometric functions which are useful, and I suppose one can say that math itself was invented to describe nature. Many have found the gaussian distribution to be useful, however, and Gauss was rather clever, so I guess it does deserve a special acknowledgement.
* Gauss had other influences.
In Maddala, G.S.(2001), Introduction to Econometrics, 3rd ed, Wiley, on page 68, he points out how important is the method of least squares by quoting from Stigler, S.M.(1981), "Gauss and the Invention of Least Squares," The Annals of Statistics, Vol. 9, No. 3, pp. 465-474.
In Applied Regression Analysis and Generalized Linear Models, 2nd ed, 2008, John Fox, Sage, page 196 and a footnote there, Fox mentions the Gauss-Markov theorem with regard to the BLUE property of least squares, but notes this extends beyond linear estimators when we assume "normality." There Fox references Rao, CR(1973), Linear statistical inference and its applications, 2nd ed, Wiley.
So the Gaussian distribution is important there. However, in practice, I haven't relied on that in regression - "normality" has not been problematic - but in some other people's work, I suppose it could be. Normality is not strictly required for estimated residuals (or the random factors of estimated residuals in weighted least squares regression), and not require for the variables at all.
Perhaps it would be best if we always said "Gaussian" rather than "normal," but that is not likely to happen now.
residuals can be roughly Gaussian even if the underlying distribution is not. This is very useful for checking model fit.
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