slides: review section 1,2,3

slides/sections/1.md

@@ -72,6 +72,7 @@ How?
 
 - Applying some hypothesis testings  
 
+
 ## Why?
 
 The Landau and Moyal PDFs are really similar. Historically, the latter distribution was utilized in
@@ -79,32 +80,31 @@ the approximation of the Landau Distribution.
 
 :::: {.columns}
 :::  {.column width=33%}
-  \centering
   ![](images/moyal-photo.jpg){height=130pt}
 :::
+
 :::  {.column width=33%}
-  \centering
   ![](images/mondau-photo.jpg){height=130pt}
 :::
+
 :::  {.column width=33%}
-  \centering
   ![](images/landau-photo.jpg){height=130pt}
 :::
 ::::
 
+
 ## Two similar distributions
 
-:::: {.columns .c}
+:::: {.columns}
 :::  {.column width=50%}
-  \begin{center}
   Landau PDF
   $$
     L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
            dt \, e^{-t \ln(t) -xt} \sin (\pi t)
   $$
 :::
+
 :::  {.column width=50%}
-  \begin{center}
   Moyal PDF
   $$
   M(x) = \frac{1}{\sqrt{2 \pi}} \exp \left[ - \frac{1}{2}
@@ -113,10 +113,11 @@ the approximation of the Landau Distribution.
 :::
 ::::
 
-:::: {.columns .c}
+:::: {.columns}
 :::  {.column width=50%}
   ![](images/landau-pdf.pdf)
 :::
+
 :::  {.column width=50%}
   ![](images/moyal-pdf.pdf)
 :::
@@ -124,5 +125,4 @@ the approximation of the Landau Distribution.
 
 ## Two similar distributions
 
-\centering
 ![](images/both-pdf.pdf)

slides/sections/2.md

@@ -3,43 +3,42 @@
 
 ## Moyal PDF
 
+
+Standard form:
 $$
-  M(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} \left[ x + e^{-x} \right]}
+  M(z) = \frac{1}{\sqrt{2 \pi}} \exp
+         \left[ - \frac{1}{2} \left( z + e^{-z} \right) \right]
 $$
-\vspace{10pt}
+
+. . .
+
+More generally:
+
+   - location parameter $\mu$
+   - scale parameter $\sigma$
+
 $$
-  \text{More generally:} \hspace{15pt}
+  z = \frac{x - \mu}{\sigma}
-  \begin{cases}
+  \thus
-    \text{location parameter:} \quad &\mu_M \\
+  M(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
-    \text{scale parameter:} \quad &\sigma_M
+         \left[ - \frac{1}{2} \left(
-  \end{cases}
+           \frac{x - \mu}{\sigma}
-$$
+           + e^{-\frac{x - \mu}{\sigma}} \right) \right]
-\vspace{10pt}
-$$
-  x \rightarrow \frac{x - \mu}{\sigma_M} \thus
-  M_{\mu \sigma_M}(x) = \frac{1}{\sqrt{2 \pi} \sigma_M}
-  e^{-\frac{1}{2} \left[ \frac{x - \mu}{\sigma_M} + e^{- \frac{x - \mu}{\sigma_M}} \right]}
 $$
 
 
 ## Moyal CDF
 
-The cumulative distribution function $F_M(x)$ can be derived from the
+The CDF $F_M(x)$ can be derived by direct integration:
-pdf $M(x)$ integrating:
 $$
-  F_M(x) = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, M(y)
+  F_M(x) = \int\limits_{- \infty}^x dy \, M(y)
   = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, e^{- \frac{y}{2}}
   e^{- \frac{1}{2} e^{-y}}
 $$
 
+. . .
 
-## Moyal CDF
+With the change of variable $z = e^{-\frac{y}{2}}/\sqrt{2}$:
-
-with the change of variable:
-$$
-  z = \frac{1}{\sqrt{2}} e^{-\frac{y}{2}}
-$$
-the CDF can be rewritten as:
 $$
   F_M(x) =
   \frac{-2 \sqrt{2}}{\sqrt{2 \pi}} \int\limits_{+ \infty}^{f(x)} dz \, e^{- z^2}
@@ -49,9 +48,9 @@ $$
 
 ## Moyal CDF
 
-given the definition of the error function `erf`:
+Remembering the error function
 $$
-  \text{erf} = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2}
+  \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x dy \, e^{-y^2},
 $$
 one finally gets:
 $$
@@ -61,37 +60,32 @@ $$
 
 ## Moyal QDF
 
-The quantile is defined as the inverse of the CDF:
+The quantile (CDF\textsuperscript{-1}) is found solving:
 $$
-  F_M(x) = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right)
+  y = 1 - \text{erf} \left( \frac{e^{- \frac{x}{2}}}{\sqrt{2}} \right)
 $$
 hence:
 $$
-  Q(x) = -2 \ln \left[ \sqrt{2} \, \text{erf}^{(-1)} (1 - F_M(x)) \right]
+  Q_M(x) = -2 \ln \left[ \sqrt{2} \, \text{erf}^{-1} (1 - F_M(x)) \right]
 $$
 
 
 ## Moyal median
 
-The median is defined as the point at which $F_M(x) = 1/2$:
+Defined by $\text{CDF}(m) = 1/2$, or $m=\text{QDF}(1/2)$.
-\vspace{15pt}
-$$
-  M(x) \thus
-  m_M = -2 \ln \left[ \sqrt{2} \,
-  \text{erf}^{(-1)} \left( \frac{1}{2} \right) \right]
-$$
-\vspace{15pt}
-$$
-  M_{\mu \sigma_M}(x) \thus
-  m_M = \mu -2 \sigma_M \ln \left[ \sqrt{2} \,
-  \text{erf}^{(-1)} \left( \frac{1}{2} \right) \right]
-$$
 
+\begin{align*}
+  M(z)
+    &\thus m_M = -2 \ln \left[ \sqrt{2} \,
+    \text{erf}^{-1} \left( \frac{1}{2} \right) \right] \\
+  M_{\mu \sigma}(x)
+    &\thus m_M = \mu -2 \sigma \ln \left[ \sqrt{2} \,
+    \text{erf}^{-1} \left( \frac{1}{2} \right) \right]
+\end{align*}
 
 ## Moyal mode
 
-Peak of the PDF
+Peak of the PDF:
-\vspace{10pt}
 $$
   \partial_x M(x) = \partial_x \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}
                     \left( x + e^{-x} \right)} \right)
@@ -99,14 +93,11 @@ $$
                     \left( x + e^{-x} \right)} \left( -\frac{1}{2} \right)
                     \left( 1 - e^{-x} \right)
 $$
-\vspace{10pt}
+
-$$
+\begin{align*}
-  \partial_x M(x) = 0 \thus \mu_M = 0
+  \partial_x M(z) = 0 &\thus \mu_M = 0                \\
-$$
+  \partial_x M_{\mu \sigma}(x) = 0 &\thus \mu_M = \mu \\
-\vspace{10pt}
+\end{align*}
-$$
-  \partial_x M_{\mu \sigma_M}(x) = 0 \thus \mu_M = \mu
-$$
 
 
 ## Moyal FWHM
@@ -115,27 +106,30 @@ We need to compute the maximum value:
 $$
   M(\mu) = \frac{1}{\sqrt{2 \pi e}} \thus M(x_{\pm}) = \frac{1}{\sqrt{8 \pi e}}
 $$
+
+. . .
+
 which leads to:
 $$
   x_{\pm} + e^{-x_{\pm}} = 1 + 2 \ln(2) \thus
   \begin{cases}
-    x_+ = a + W_0 \left( - \frac{1}{4 e} \right)    \\
+    x_+ = 1 + 2 \ln(2) + W_0 \left( - \frac{1}{4 e} \right)    \\
-    x_- = a + W_{-1} \left( - \frac{1}{4 e} \right)
+    x_- = 1 + 2 \ln(2) + W_{-1} \left( - \frac{1}{4 e} \right)
   \end{cases}
 $$
-with $a =  1 + 2 \ln(2)$
-
 
 ## Moyal FWHM
 
 $$
-  \text{FWHM}_L = x_+ - x_- = 3.590806098...
+  x_+ - x_- = W_0 \left( - \frac{1}{4 e} \right)
-$$
+          - W_{-1} \left( - \frac{1}{4 e} \right)
-\vspace{15pt}
+          = 3.590806098...
-$$
+          = a
-  M(x) \thus \text{FWHM}_M = 3.590806098...
-$$
-\vspace{15pt}
-$$
-  M_{\mu \sigma_M}(x) \thus \text{FWHM}_M = \sigma_M \cdot 3.590806098...
 $$
+
+\begin{align*}
+  M(z)
+    &\thus \text{FWHM}_M = a              \\
+  M_{\mu \sigma}(x)
+    &\thus \text{FWHM}_M = \sigma \cdot a \\
+\end{align*}

slides/sections/3.md

@@ -3,12 +3,19 @@
 ## Data sample
 
 The $M(x)$ most similar to $L(x)$ is found by imposing:
-\vspace{15pt}
+
+- equal mode
 $$
-  \mu_M = \mu_L \sim = −0.22278298...
+  \mu_M = M_L \approx −0.22278298...
 $$
-\vspace{15pt}
+
+- equal width
 $$
-  \text{FWHM}_M = \text{FWHM}_L = 4.0186457...
+  \text{FWHM}_M = \text{FWHM}_L = \sigma \cdot a
-  \thus \sigma_M = 1.1191486
+$$
+
+. . .
+
+$$
+  \implies \sigma_M \approx 1.1191486
 $$