sections: write a lot

slides/sections/3.md

@@ -66,7 +66,7 @@ $$
 $$
 hence:
 $$
-  Q_M(x) = -2 \ln \left[ \sqrt{2} \, \text{erf}^{-1} (1 - F_M(x)) \right]
+  Q_M(y) = -2 \ln \left[ \sqrt{2} \, \text{erf}^{-1} (1 - y) \right]
 $$
 
 

slides/sections/4.md

@@ -3,7 +3,8 @@
 
 ## Sample parameters estimation
 
-Once the points are sampled, how to estimate their median, mode and FWHM?
+Once the points are sampled,  
+how to estimate their median, mode and FWHM?
 
 . . .
 

slides/sections/5.md

@@ -14,7 +14,7 @@ FWHM               $w_L\ex$          $w_M\ex (σ)$
 -----------------------------------------------------
 
 
-## PDF parameters
+## Moyal parameters
 
 A $M(x)$ similar to $L(x)$ can be found by imposing:
 
@@ -37,7 +37,7 @@ $$
 $$
 
 
-## PDF parameters
+## Moyal parameters
 
 :::: {.columns}
 :::  {.column width=50%}
@@ -50,7 +50,7 @@ $$
 ::::
 
 
-## Different medians
+## Moyal parameters
 
 This leads to more different medians:
 
@@ -60,7 +60,22 @@ This leads to more different medians:
 \end{align*}
 
 
-## Samples
+## Results compatibility
 
-- Sample $L$: N = 50'000 points following $L_(x)$
+Comparing results:
-- Sample $M$: N = 50'000 points following $M_{\mu \sigma}(x)$
+
+$$
+  p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
+  t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
+$$
+
+- $x\ex$ and $x\ob$ are the expected and observed values
+- $\sigma_e$ and $\sigma_o$ are their absolute errors
+
+. . .
+
+At 95% confidence level, the values are compatible if:
+
+$$
+  p > 0.05
+$$

slides/sections/6.md

@@ -0,0 +1,65 @@
+# Landau sample
+
+## Sample
+
+Sample N = 50'000 random points following $L(x)$
+$$
+  L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
+         dt \, e^{-t \ln(t) -xt} \sin (\pi t)
+$$
+
+
+## Compatiblity results:
+
+Median:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $t = 0.761$
+  - $p = 0.446$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Compatible!}
+  $$
+:::
+::::
+
+\vspace{10pt}
+
+. . .
+
+Mode:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $t = 1.012$
+  - $p = 0.311$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Compatible!}
+  $$
+:::
+::::
+
+\vspace{10pt}
+
+. . .
+
+FWHM:
+
+:::: {.columns}
+:::  {.column width=50%}
+  - $t=0.495$
+  - $p=0.620$
+:::
+
+:::  {.column width=50%}
+  $$
+  \thus \text{Compatible!}
+  $$
+:::
+::::