From 8eb4857d9a3d05825e8d96f396526d7de4ae81fd Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Fri, 5 Jun 2020 23:30:39 +0200
Subject: [PATCH] slides: move things from notes-moyal.md to the sections

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 slides/misc/notes-moyal.md | 34 ++++++++++++++++++++++++++++++++++
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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.