# The Analysis of WJ Distribution as an Extended Gaussian Function: Case Study
###### Abstract  
The double exponential WJ distribution has been shown to competently describe extreme events and critical phenomena, while the Gaussian function has celebrated rich applications in many other fields. Here we present the analysis that the WJ distribution may be properly treated as an extended Gaussian function. Based on the Taylor expansion, we propose three methods to formulate the WJ distribution in the form of Gaussian functions, with Method I and Method III being accurate and self-consistent, and elaborate the relationship among the parameters of the functions. Moreover, we derive the parameter scaling formula of the WJ distribution to express a general Gaussian function, with the work illustrated by a classical case of extreme events and critical phenomena and application to topical medical image processing to prove the effectiveness of the WJ distribution rather than the Gaussian function. Our results sturdily advocate that the WJ distribution can elegantly represent a Gaussian function of arbitrary parameters, whereas the latter usually is not able to satisfactorily describe the former except for specific parameter sets. Thus, it is conclusive that the WJ distribution offers applicability in extreme events and critical phenomena as well as processes describable by the Gaussian function, namely, implying plausibly a unifying approach to the pertinent data processing of those quite distinct areas and establishing a link between relevant extreme value theories and Gaussian processes. distribution function; applied physics; gaussian function; WJ distribution; scaling; critical phenomena +
Footnote †: journal: Medical Image Processing  
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Footnote 1: email: {[[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
thermodynamic and diffusion equations [10; 11], define Gaussian light beams [12] and have superior niches as the wave function describing the ground state of a quantum harmonic oscillator, as molecular orbitals through their proper linear combinations and being related to the vacuum state in quantum field theory. Medical phenomena such as mass population heights, red blood cell numbers, and hemoglobin amounts are adequate to be characterized or approximated by Gaussian distributions, with multidimensional-multivariate generalizations in perspective. To a certain extent, researches in educational statistics confirm that students' academic performance and practical ability seem to follow Gaussian propensities. In reality, the Gaussian distribution is a milestone in the development history of statistical theory, with the probability density function as  
\[f(x,\mu_{GS},\sigma)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu_{GS})^{2}}{2 \sigma^{2}}}\] (1a) where \[\sigma\] and \[\mu_{GS}\] are the respective standard deviation and expectation of a random process, defined over the domain of \[(-\infty,+\infty)\]. The commonly used form of Equation ( 1a ) is known as the standard normal distribution when \[\mu_{GS}=0\] and \[\sigma=1\], simply stated as \[f(x,\mu_{GS}=0,\sigma=1)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}} \tag{1b}\]  
It appears relevant to emphasize that the parameters \(\sigma\) and \(\mu_{GS}\) are independent of each other, i.e., the former affects solely the shape of the curve while the latter exclusively determines the location of its center (which is markedly different from the WJ distribution as discussed below [13]. As a note for the WJ distribution, we have wanted a short term to represent the distribution for the simplicity of discussion and come up with the abbreviation WJ which could mean wild, wide, or weird, even wonder jittering to reflect unusual occurrences such as extreme events and critical phenomena).