From a84529f2d41099957dd3736761a48db6cc2a5ca7 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Sat, 6 Jun 2020 19:38:57 +0200
Subject: [PATCH] misc: rename notes_moyal.md -> notes-moyal.md

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-The Moyal distribution, which  is a steepest descent approximation of the
-Landau distribition, is defines as:
-$$
-  \exp \left( - \frac{x - \mu }{2 \sigma}
-              - \frac{1}{2} \exp \left( - \frac{x -\mu}{\sigma} \right) \right)
-$$
-Mean $m$ and variance $\sigma$:
-$$
-  m = \mu + \sigma [ \gamma + \ln(2) ] \et \sigma = \frac{\pi^2 \sigma^2}{2}
-$$
-Median:
-$$
-  \mu - \sigma \left[ 2 \text{erf}^{-1} \left( \frac{1}{2} \right)^2 \right]
-$$
-skewness and kurtosis are constant:
-$$
-  s = \frac{28 \sqrt{2} Z(3)]{\pi^3} \et k = 7
-$$
-max value:
-$$
-  \frac{1}{\sqrt{2 e \pi}}
-$$
-cdf:
-$$
-  \text{erf} \left( \frac{\exp \left( 
-  - \frac{x - \mu}{2 \sigma} \right)}{\sqrt{2}} \right)
-$$
-
-$\mu$ is the location parameter and $\sigma$ is the scale parameter.  
-The Moyal distribution was first proposed in a 1955 paper by physicist J. E.
-Moyal. The distribution models the energy lost by a fast charged particle
-(and hence the number of ion pairs produced) during ionization. Historically,
-the Moyal distribution has been utilized in the approximation of the Landau
-Distribution and has since found use in modeling a wide array of phenomena.
-
-
-
-# PDF
-
-The Moyal distribution is defined as:
-$$
-  M(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} \left[ x + e^{-x} \right]}
-$$
-More generally, it is defined with the location and scale parameters $\mu$ and
-$\sigma$ such as:
-$$
-  x \rightarrow \frac{x - \mu}{\sigma}
-$$
-
-# CDF
-
-The cumulative distribution function $\mathscr{M}(x)$ can be derived from the
-pdf $M(x)$ integrating:
-$$
-  \mathscr{M}(x) = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, M(y)
-  = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, e^{- \frac{1}{2}}
-    e^{- \frac{1}{2} e^{-y}}
-$$
-with the change of variable:
-\begin{align}
-  z = \frac{1}{\sqrt{2}} e^{-\frac{y}{2}}
-  &\thus \frac{dz}{dy} = \frac{-1}{2 \sqrt{2}} e^{-\frac{y}{2}} \\
-  &\thus dy = -2 \sqrt{2} e^{\frac{y}{2}} dz
-\end{align}
-hence, the limits of the integral become:
-\begin{align}
-  y \rightarrow - \infty &\thus z \rightarrow + \infty \\
-  y = x &\thus z =  \frac{1}{\sqrt{2}} e^{-\frac{x}{2}} = f(x)
-\end{align}
-and the CDF can be rewritten as:
-$$
-  \mathscr{M}(x) = \frac{1}{2 \pi} \int\limits_{+ \infty}^{f(x)}
-  dz \, (- 2 \sqrt{2}) e^{\frac{y}{2}} e^{- \frac{y}{2}} e^{- z^2}
-  = \frac{-2 \sqrt{2}}{\sqrt{2 \pi}} \int\limits_{+ \infty}^{f(x)}
-  dz e^{- z^2}
-$$
-since the `erf` is defines as:
-$$
-  \text{erf} = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2}
-$$
-$$
-  1 = \frac{2}{\sqrt{\pi}} \int_0^{+ \infty} dy \, e^{-y^2}
-    = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2} + 
-      \frac{2}{\sqrt{\pi}} \int_x^{+ \infty} dy \, e^{-y^2}
-    = \text{erf}(x) + \frac{2}{\sqrt{\pi}} \int_x^{+ \infty} dy \, e^{-y^2}
-$$
-thus:
-$$
-  \frac{2}{\sqrt{\pi}} \int_x^{+ \infty} dy \, e^{-y^2}
-  1 = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2} + 
-$$