From cf785830e2de20ac9df989d14fca27d0a28f9008 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Thu, 4 Jun 2020 21:46:27 +0200
Subject: [PATCH] pres: create the file counts.md and write something

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 pres/counts.md | 43 +++++++++++++++++++++++++++++++++++++++++++
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 create mode 100644 pres/counts.md

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+# 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:
+$$
+  
+$$